I think despite what a lot of people say about any sort of allegations that went on around the time of Sonic 3's development, I think one area that people should explore more into is contract law.
Forget about Michael Jackson for a moment, and think about the terms and conditions that both parties (Sega and the song writing team) potentially agreed upon and wrote on paper. You have to remember that back in 1993, the concept of having music licensing involved with video games was quite rare. The other thing is that you have to realize that Sega as a company probably wasn't as forward thinking about the games they made when opportunities would eventually come about to rerelease games. Understanding the terms of the contract that was negotiated and signed between Sega and the artists who wrote the songs is the key to understanding the entire situation, from the decisions made in the past to the current predicament today.
This is my hypothesis of what occurred back then and how and why it applies even today.
The main issue that persists from the beginning up to now, in my opinion, has always been about ownership rights. One thing out of all of this that I'm 99% sure of is that besides the contract work done within Japan (between SoJ and CUBE/OPUS corp) and SoJ's own sound team, is that Sega does NOT own exclusive rights to the contract work given by Brad Buxer and his sound team, and they never did. When Sonic 3 was in development, a written contractual agreement between Sega and Brad Buxer's song writing crew was most likely made that gave Sega the rights to use and/or interpret the works that Brad Buxer and his team provided with permission (either it was SoA or SoJ is anyone's guess, but I'd imagine this was a SoA thing since MJ visited STI as well during this time). What the terms of the agreement were are completely unknown, but given that legal litigation occurred a decade ago about possible owed royalties, I would wager that the contract guaranteed royalties for whatever copies of "Sonic the Hedgehog 3" were sold for the "Sega Genesis video game/entertainment system". The terms and conditions of the contract didn't give Sega any exclusive ownership of any of the work contracted by this specific crew, so this would've restricted Sega to what they could/couldn't do with some of the music that they were given. For instance, Sega didn't have the rights to sell a soundtrack containing Brad Buxer's music, they couldn't use it in marketing, and they would have to negotiate with the rights holders every time the music was to be used with permission. It's not known if these rights were meant to expire eventually either. Compare this to the contract work done through OPUS/Cube, or Sega of Japan's own sound team, where they were able to negotiate full ownership of all the works created.
I think the decision to use Brad's tracks in the first place was a creative one rather than a purely marketing decision. If Sonic CD is anything to go by, it's possible that the tracks were chosen because someone thought they were better, at least for an American audience, which was always the thing with Sonic in particular since he was made pretty much for an American audience in mind from the very beginning.
However, this kind of complicated things as now Sega only really legally owned the rights to SOME of the music in Sonic 3, not everything or nothing. Obviously any work done under employment with Sega themselves meant that any work Howard Drossin and the Sega Sound Team would do would be completely owned by Sega. But the work Brad Buxer and his crew did would be the exception, which would eventually become a headache I think Sega realized would've been an issue right from the start.
When development of the first half of Sonic 3 wrapped up and work began on the second half, Sega had no issue using some of the music Brad Buxer produced in Sonic 3 and selling it that way. But I think Sega realized at the time that they were missing out on a larger piece of the profit pie by giving some of it to Brad Buxer's writing crew. So it was decided that Sonic & Knuckles would be treated as a brand new game (at least by name) and would contain zero music from the contractual agreement that Sega had made with them. This meant - a new title theme, new title screen, new Knuckles theme etc. I think it might've been possible that there was a brief moment where maybe Sega wanted to continue using Brad Buxer's tracks for Sonic & Knuckles, but the idea might've been abandoned for various reasons from every party involved (either it be the sound quality thing, the MJ allegation thing, or whatever, take your pick or all the above). This way, Sonic & Knuckles would be a product that Sega owned the rights to in every regard. The lock on mechanic was an additional bonus where they could theoretically still use the Brad Buxer tracks when combined with a purchased copy of Sonic 3, as long as Sonic & Knuckles standalone didn't contain Brad Buxer's compositions. This also gave Sega an opportunity to sell Sonic & Knuckles' soundtrack on CD, since they actually owned all the tracks on the CD, which they wouldn't have been able to do with Sonic 3.
So Sonic 3 and Knuckles came and went, and it would be around 3 years or so before the game would resurface again. The next release of Sonic 3 came in the form of the Sonic & Knuckles Collection for the PC in 1997. For this release, Sega decided to use the original compositions that can be found in Origins. The decision to use the tracks could've been possibly a combination of legal reasons and technical reasons (since they wanted to use MIDI rather than prerecorded music). The contract that Sega agreed to with Brad Buxer's song writing crew possibly only allowed the use of the songs in the original Mega Drive version only, or it's possible that Sega of Japan just wasn't interested in paying royalties again. This would also explain those specific tracks' absence from the Screensaver part of the collection, as only those Brad Buxer songs are missing from the sound selection. Sonic Jam would be released later that same year with all of Brad Buxer's songs intact, but who knows if there was another negotiation for this release or if the contract allowed usage of the music as long as it was meant for a video game console, whereas the PC collection might've been considered a different distinction legally (PC/computer software vs entertainment).
Sonic 3 wouldn't see another release until the Sonic Mega Collection in 2002. One thing you have to remember with Sega (especially Sega of America), the people/higher ups who worked for the company weren't entirely the same every few years or so. The Sega of 1989 was different from the Sega of 1994, which was different from the Sega of 1997, which was different from the Sega of 2002, 2006, etc. I think as time went by, the idea that there were special conditions for any rerelease of Sonic 3 was forgotten about as the old guard left Sega and the new guard came in. Sega went through a few corporate structural changes throughout the 2000s, the first of which began sometime in 2001/2002 when the company became a third party developer for other publishers. When a company restructures itself, a lot of things get lost in the change. Aside from employees, knowledge of prexisting things like company assets (legal agreements, source code, documents, etc) that aren't thought to be necessary going forward at the time are quickly discarded. It's possible that whatever knowledge of any existing contractual agreements with past software was just simply lost or thrown away, and the new staff that came to replace the old weren't aware of it.
And so Sonic 3 continued to be rereleased quite a few times throughout the 2000s without incident. At the same time, those who were contracted to work on the soundtrack for the game probably forgot about it or didn't think too highly of it to reconsider (I think if your career rode on the success of Michael Jackson, I don't think you'd give two shits about Sonic the Hedgehog).
However, the issue in regards to music ownership and rights would come to a head once again when interest in the identity behind Sonic 3's soundtrack became a huge interest. Suddenly, people started caring A LOT about Sonic 3's soundtrack and the fact that Michael Jackson of all people were possibly behind it. This revelation caused people to reach out to the composers who worked on the game asking for clarification on the work that they contributed to the game. Before this happened, I don't think many really cared about their contributions to the project. To Brad Buxer and crew, Sonic 3 was probably thought to be a "once and done" project, quickly forgotten before moving on to another profitable project. But the sudden attention was probably something that spurred their decision to look into what the hubbub was all about. It was probably at this time when people began contacting Brad Buxer and the song writing crew that they suddenly realized that Sega was still selling Sonic 3 in making bank off of the Virtual Console, DS, console, and Steam rereleases of the game as well as possible past compilations. The only issue was that, possibly, this went against the terms of the contract that was originally written in 1993 between the artists and Sega. This was money that I think the song writing crew felt was owed, and if the law favors the song writers' rights, they are owed quite a LOT of money.
This was the reason Sonic 3 wasn't really acknowledged starting with the 2010s on. It's possible that negotiations were held on and off throughout the years to try and get the legal clearing to use the music once again. However, Sega owes people a lot of money and they aren't exactly sure what the original contract might have entailed, and the song writers know their rights as anyone reasonably should. Music rights is still a gigantic issue to this day, and was a big issue even back then, so most pop song writers (especially the big ones) KNOW their rights and know what they own. This is a game that every party except Sega is familiar with.
I think Sega's ultimate goal, besides succeeding in a renegotiating a new contract with the song writers, would be to eventually own all of Brad Buxer and his team's contributions. It's not known if the option ever became available to Sega at some point, but I'm sure that if it did it would have costed Sega a lot more money than they would've liked. Also note that this wouldn't clear Sega of any past discrepancy or unpaid royalties, so those would have to be cleared before any renegotiations can occur. But good luck trying to do book keeping on 20+ years of rereleases and sales! But it's also possible that Brad Buxer and his song writer's aren't interested in selling, and why would they? If Sega would've honored whatever contract they signed from almost 30 years ago, not only would the crew gain past royalties, but they would also be clued in on more royalties for the 1000 eventual future rereleases of Sonic 3. So outright selling the rights to Sega probably wouldn't be in anyone's interests, especially if royalties seem more lucrative by comparison.
Unfortunately, everything I say is hypothetical because I don't think there's public access concrete legal information about any negotiations that occurred. I wish Brad Buxer would talk more about what exactly the negotiations were and what he thinks his rights are, but as far as I know I'm not too sure if anyone has made a public comment about it. But until then, who knows what really is the truth.