OP, I would recommend trying to reach out directly for referrals on LinkedIn instead if you are not already. That will at least make it likely your resume will get looked at. It is correct that you have a visibility problem, not a skills one.
To get a PR in you have to be pretty actively using the project and be lucky enough to spot an issue worth fixing. Either that or you're going through Issues trying to find something important enough that someone isn't working on.
Let's say you find the thing, you fork the repo (because you aren't on the team) make the change, open the PR, then it's back to Discord trying to get someone to look at the PR. Sometimes a miracle happens and they approve the PR, but you still aren't allowed to merge it, gotta usually bother them one last time to merge it.
Worth it if it's a project you use all the time and you truly understand the codebase, I do this all the time with stuff I use, but nearly impossible if you're an outsider just trying to get a PR merged on something you don't use especially in a well-known repo.
I've done that a few times for jobs i was particularly excited to apply for, but didn't think about it from the perspective you mentioned. I'll definitely do that from now on.