I always viewed the suggestion spell as doing something the NPC might be inclined to do already. I was never a fan of "me cast cantrip-level spell get god-tier results" thing.
It turns out that there's a lot of debate on what exactly you can get away with with suggestion, helped in no small part by WotC not really deciding to do any major clarifications beyond "let the DM decide." That really doesn't help address the intended limitations.
Part of my argument against the player was that if suggestion were as powerful as he implied, there would be no need for stronger charm spells because you could just suggest the big bad to hand over the McGuffin and call it a day, the existence of stronger spells implying that lower level spells couldn't do as much. But nope, he continued to insist that as long as you weren't actively trying to tell someone to kill himself, you'd be just fine.
Leave to the DM, of course. FWIW, I've had similar situations come up, and my answer is always, "that's more appropriate for Dominate Person." The knight in the RAW example presumably does not immediately demand her horse back (IMO, it's also a terrible example, as the soy-fueled types who wrote the rules probably do not have any idea just how expensive a horse actually is). A reasonable request in this situation might be, "Give the ring for safekeeping - this area is dangerous, and if you get eaten by a grue, it could easily fall into the wrong hands. I'm quite a bit better at handling adversity than you, as you have seen."
See, with regards to the PHB example, in theory you could potentially make it work, perhaps by playing on the knight's sense of honor and chivalry ("the poor are in dire need of your horse, ride off and give it to the first you've seen"). It wouldn't be something likely they'd do, again considering how important it would be to them, but in a time of desperate need, they might part ways with it. This was another point in our discussion, where I argued that "you really don't want to fight us and die, get out of here" was a reasonable suggestion that appeals to one's self-preservation instincts, but he replied that by my interpretation, it wasn't reasonable at all to abandon a fight you're in the middle of.
Ironically, your formulation of a suggestion would be
terrible for him to try. Thus far, the NPC has been a hell of a lot more useful than the wizard, which is part of why my bard was pissed at him. At the start of combat with the Red Wizards, we were both at the top of the initiative order, with him going first. All the wizards were clustered together, it was the perfect opportunity to drop an AoE. So what does he do? Wastes his first turn flying away, then when I try to hit them with Hunger of Hadar and get counterspelled, he doesn't counter the counter (probably wouldn't have worked anyway since that counter would be countered by a different wizard, but it wouldn't have hurt). He's not completely useless, but everyone else does a lot more while he's safely hovering above the battlefield, and if things had started looking grim he would 100% have bailed. (The irony that he said that abandoning a fight was not reasonable is not lost on me.)
He's already committed to ditching this character one way or the other, I just think it'd be stupid to let him waltz off with an artifact due to a rules interpretation that might be RAW, but certainly doesn't feel like RAI.
I feel this would be a wonderful way to start the next PC questline, centered around the bard - but I'm unsure how to proceed. How can the bard get his literal groove back?
The bard has now become a Slam Poet.
I think this is one of those teaching moments.
I like this idea myself. Nothing says a bard has to specifically play music to have their spells work, and it certainly makes for an interesting story. And it was Vecna that took his music? Good luck getting that back until much later (I'm assuming the party isn't very high level yet).
That said, if you wanted to work this into a questline, perhaps it's not literally Vecna who created the portrait, but another powerful wizard that used Vecna's notoriety to make the painting more valuable. The questline could involve researching the painting and its origins, then tracking down the wizard in his lair and finding a way to retrieve the bard's musical ability. The climactic showdown has the bard finally regaining his lost hope and belting out a power ballad that melts the wizard's face off.
And you can always go with a cheesy "the music was inside you all along" reveal at the end if you want, but that's up to you.