They exist, and they aren't that rare. My org currently has a noticeable percentage of staff+ ICs, not managing, and I'm actively encouraging folks to think careful if they really want management. I am far from the only manager doing that.

But yes, jobs are very rare. Staff+ ICs tend to like to stay where they work (because great achievements take time), there's only a limited number of roles, and a lot of this is word-of-mouth in a given field.

I'd look either at FAANG - they're large enough to always need staff+ folk - or startups once they have round A. They live or die by having really experienced and productive staff+ folks, and they're large enough that they can offer staff+ interesting challenges.

Why not pre-founding? At that point, any good staff engineer I know is driven enough that they'd be better off just starting their own. In a small startup, the founder is either technical and interested in solving the hard questions, or non-technical and you'll have essentially a CTO role, but without founder credit.

It's not a hard-and-fast rule, there are exceptions to everything - but that's where I'd focus my search if I were to search for that kind of role.