I wish that was true, there are so many games I would go back and play again. PS3 was a technical marvel for it's time, however using it was so complicated that we only got to see it's true potential by the end of it's lifecycle with games like TLOU. So just speaking of raw power, PS5 could %100 not emulate PS3 games flawlessly as we barely have PS3 emulation on PCs. Technical part aside, it just doesn't make business sense for Sony to invest in it. You also have to keep in mind if Sony offers PS3 emulation, it can't stutter or crash on some games. it needs to be a commercial grade product which is not even guaranteed with PS2 or NES emulation on PC
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I've played lot of games on the PC without issue like lollipop chainsaw, asura's wrath, drakengard 3, and shadows of the damned without issue at 4k/60. Was really surprising to me, but rpcs3 has made great strides and if a game has been marked playable it's been a fantastic experience. And that's with someone unaffiliated with Sony making it possible. Seeing what Xbox accomplished with Xbox One when it came to 360 compatability it wouldn't surprise me if Sony did better than what RPCS3 did.
There's some notable attention given the games that don't emulate well like MGS4 and red dead with it coming across as buggy, but lot of other titles do run really well as if they were official remastered ports.
This is what I am saying. They don't have to offer the whole back catalog. But they can at least try and offer what is currently working in emulation.
Xbox has always doing the x86 architecture(Edit: as corrected by following comment, 360 was not.) so it's much easier to do BC. For Sony or Nintendo it's just not worth the effort until the emu is mature that they can just reap the benefit. PS5 can already play almost entire PS4 library, Anything PS2 or before can be emu pretty consistently if you are trying to get it done, then it only left PS3 in a weird place. For PS3, many good games already have a PS4/PS5 remaster games, for non-best sellers you can probably get a cheap ps3 slim with enough storage to play those left out games.(ie, PSN only Puppeteer), OR stream them like you mentioned.
This interview has some details about how it was done on Xbox: https://hanselminutes.com/632/eric-heutchy-on-backward-compatibility-xbox-one-x-enhanced-games-and-the-heutchy-method
Basically all games of note going all the way back to the OG Xbox