4e and PF2 share the same genetics. They're both reactions (at least to an extent) to 3.x and the issues inherent to that edition. So it isn't terribly surprising that they ended up with somewhat similar solutions when they were solving the same 'problems'.
As for why the game isn't more like 5e, I think they wanted to make their own thing rather than chasing after WotC. Which is perhaps a bit ambitious but commendable in it's own way. I realize that some people really just wanted a more rules heavy variant of 5e, but my impression is that that's not the direction that the designers wanted to go.
Is it better to design the game you envision or the game that is successful? (I mean, in an ideal world you probably want both, but this is assuming you can only have one or the other.) The answer is obviously subjective. I think that they went with the game that they wanted to design and hoped it would be successful (rather than trying to design a game for success but not having it be a game they wanted to create).
Given that we don't know what their sales goals were for the game, and we have only limited data on how well the game is doing, all we can really do is speculate as to whether or not the game has been successful. Heck, the Gamemastery Guide is just coming out now, so I think it's a bit premature to make that determination. Is it a 5e killer? Definitely not. But I don't think they ever set out to do that, so unless I'm wrong, that was never a metric for the success of PF2. Ultimately, only Paizo really knows whether or not the game met their expectations.
Or just special in a different way, or good at a different thing.
Comes back to strong niche protection (strengths) and significant weaknesses being good things. I'm really good at this, you're really good at that, she's really good at something else, and together we cover off each others' weaknesses except we're all awful at the fourth thing; we'd better go recruit someone who can do that for us.
In fairness though, 4e did have fairly strong niche protection in its roles. A striker couldn't really do a defender's job, and neither of them could replace the leader. There were also more specialized applications by class. The cleric and bard were both leaders, but the cleric was far more capable of granting their support from the front line, whereas a bard would need to hang back more.
I get that it wasn't to everyone's taste, and that's fine.
However, when I see "if everyone is special then no one is" it really seems to smack a bit of "if everyone gets to do cool stuff then no one does". Which is something I just can't wrap my head around. Bob the fighter getting awesome moves that he can use doesn't diminish my wizard in the least, IMO. Niche protection is important in class based games, but no one's niche should ever be 'all the cool stuff'.