Don't some SSDs have 512b page size?

I would guess by now none have that internally. As a rule of thumb every major flash density increase (SLC, TLC, QLC) also tended to double internal page size. There were also internal transfer performance reasons for large sizes. Low level 16k-64k flash "pages" are common, and sometimes with even larger stripes of pages due to the internal firmware sw/hw design.

Also due to error correction issues. Flash is notoriously unreliable, so you get bit errors _all the time_ (correcting errors is absolutely routine). And you can make more efficient error-correcting codes if you are using larger blocks. This is why HDDs went from 512 to 4096 byte blocks as well.

They might present 512 blocks to host, but internally the ssd almost certainly manages data in larger pages

And the filesystem will also likely be 4k block size.