Well holy crap, a game journalist actually taking Sony to task over legacy BC. I need to update my list of times this has happened to... let me see, nope, this is the first. Not counting Time magazine accidentally getting the quote of the year out of Jim Ryan since they're not a gaming focused publication.
It really doesn't as you can't play PS2 games you previously purchased with it, you can only play the ones bought for the PS4. So if you have discs or PS2 Classics for the PS3, it doesn't matter. So from the point of view of "Does Sony have a PS2 emulator?" the answer is yes, but "Does PS4 have PS2 BC?" it's no.
Just to give you an idea of how bad things are, the Xbox One and One X have a dev mode, where for $20 you can install third-party unsigned and open-source software. This includes a port of Retroarch meant specifically for Xbox. Presumably the XSX will retain that. Retroarch includes excellent emulators for the PS1 and PSP, as well as an okay one for PS2 (Play!, not PCSX2). And it's possible someone might port PCSX2 and RPCS3 to the Xbox app.
With an XSX in dev mode you could play pretty much every PS1 and PSP game, as well as some PS2 games. With some real work, possibly even most PS2 games and fair number of PS3 games. Unlike the PS5.