That'll only be a problem if you're up against significant number of candidates that have both experience and a PhD. HR will certainly filter resumes based on something as clear-cut as a degree when they can. But they can't just filter everything out and tell the hiring manager he's SOL. So you don't have to check all the boxes as long as you check enough of them, and you're competitive with the other candidates.

At the same time, nobody is going to do all that well if they just apply online and cross their fingers. You need some kind of human contact, either though an introduction to an insider through your network, or through a recruiter of some kind. It takes a bit of time to develop the relationships, but it's quite doable and worth it, even for introverts. Best to start before you're interested in changing jobs.