Recently, Sony PlayStation announced the new PS Plus Premium pack which allows gamers to play PS1, PS2 and PSP games with the ability to download them directly to the PS4/PS5 consoles, however, Sony confirmed that PS3 games will still use cloud-based streaming. 
Sony announced in 2021 that they would close the storefronts for PlayStation 3, PlayStation Portable, and PlayStation Vita games, as a result of the negative feedback, Sony announced on April 19, 2021, that they had reversed their decision to close the PS3 and Vita stores (https://blog.playstation.com/2021/04/19/playstation-store-on-ps3-and-ps-vita-will-continue-operations/ 

Currently, the only solutions to play PS3 games outside its console, are via PS Now cloud streaming or the community-based emulation called RPCS3. As gamers, we know that the stores will not be open forever. We were hoping that Sony was preparing to come up with a solution to fix this problem with their intentions to preserve video game history.

Based on experiences, many PlayStation customers have pointed out that using PS3 cloud-based streaming is not reliable, and requires a consistent, reliable internet connection which not all regions around the world have.

A better solution would be to provide gamers with PS3 emulation built by Sony and included in PS5 via a software update and the ability to use our PS3 game discs to either boot or download the latest versions of selected PS3 games which would download to the console, much like Microsoft's backwards compatible solution.