Are you sure about that? I have a personal mental exercise / thought experiment where I try to iterate on my understanding of what it means to exist. But I wouldn't call that foundational. I maybe run through it once a month or so. Certainly it's attained a great deal of importance, but really, iterating on my understanding is never going to really get me anywhere.

Anything can be said to "exist". If you are thinking about it, it's currently existing in your mind as thought. If you're not thinking about it, but have thought about it in the past, then it exists in your memory. If you've never thought about it, then it's a potential that you could experience someday. It could also exist as a potential quantum state that nothing living could ever consider.

I'm reminded of the "library" where every possible grouping of characters is tied together with a math function and you can search for whatever you want. Considering whether everything ever written 'exists' or not just kinda lost its thunder to me. If everything meets some criteria, then it's the criteria that needs to be examined, not the everything. Archimedes wanted to move the world with his lever, but really, it's way more interesting to just move half.

You'd have to somehow escape your own mind before you could truly consider something that doesn't exist somewhere.

I don't think existence, in that sense, is as big of an idea as people seem to think. The way I usually use it is to get at the fundamental quality of being human. I think about agency, goals, limitations, purpose. These things seem way more interesting than trying to drive at the fundamentals of everything.

What it means for me to exist is way cooler to think about than what it means for my bed to exist.