> I haven't got a PhD but I can (at least) learn and think on that level.

How do you know this? More pertinently, how do I know this? How can you convince others that you are "at PhD level"?

Well, one way to do this is to do something "at PhD level". Build something which (i) is appealing to the kind of people whom you would like to have as colleagues, and (ii) shows your mastery of the full stack (as you define it). Put it up on GitHub, and let others see it. Show them the code, as the saying goes.

What would such a thing be? Knowing what the "interesting" problems are is part of "being at PhD level". So you probably already know what to solve, or it will come to you if you think about it a bit.

Good luck with your efforts!