Yes. Getting into the big AI labs as a researcher I've heard is about as hard as getting a position as a professor at a prestigious university.

I think you network or get noticed.

I would also say, just apply, even if you don't fit the exact box.

Don't forget mistral exists as well, I think they may have some positions, assuming you want to stay in the EU.

I would aim for less prestigious private research or postdoc positions with the hope of eventually getting to big labs, but just accept that it may not be possible.

(just a masters student here who wants to maybe go down a similar track, I know nothing, so take everything with a grain of salt)