Looking at what you want, I could give my two cents on it
- Proper emulation for PS3 games
- Balance of performance and graphics
...
- Multitasking for web browsing
- Would that be possible with a budget of $1200-1500
For these kinds of use cases, the Steam Machine would seem like a good choice. It may cost less than the usual PC (especially if you compare it to something like the Framework Desktop, which is more powerful but also potentially 3x as expensive). PS3 emulation has been conquered with the addition of AVX-512 on Zen 4 and 5, which the Steam Machine will definitely feature. However...
- On range with Xbox Series X to ponder playing newer games
The Steam Machine is not very upgradable. Yes, you can change the SSD and RAM sticks, but you will be stuck with a (pretty good) laptop-tier GPU with 8GB of VRAM. It will probably be good enough for all of your games, but PCs are made to be upgraded.
If I were to purchase a new computer, I would divide the build in three purchases:
- Try to look for a combo deal for RAM. Online dealers may bundle a power supply, motherboard and other stuff in order to prevent scalpers from buying everything.
- You can purchase a used GPU on eBay, if you know where to look for. AMD 7000 GPUs can be bought at a reasonable price.
- Get the other components as needed.
I did spend around $1500 on a new build, however you have to consider I already owned a GPU and power supply, plus it was before the huge price hike on chips.
If the rumors are true, the Steam Machine will release in February/March. It may not be the best computer for $1000, but it would be fair enough for most of your games.