The reality is that yes, being full-pay helps. They won’t take you if you aren’t qualified to go to that school, but if you’re the same as someone who needs financial aid, they’ll probably pick you. This question is asked at every FA info session I’ve been to, and while they always say “admissions is a different department” and “we won’t accept an unqualified student” they also say that they have a fixed amount of cash to spend on FA, and if they don’t give it to you they can give it to someone who needs it. So, in essence, they choose to accept a certain number of full-pay students.
I’m not talking about just state schools, either – we visited some top LACs with enormous endowments. As one of the them said, “No matter how much money we take in, we spend it. The college has professors to pay, lights to keep on, dorms & research labs to maintain. So admissions does tend to put the students not applying for financial aid in a separate pile – even if it’s just in their minds. They hate it, but they do it.” (may have a few words wrong but that’s pretty much the quote - I remember this clearly)
As someone whose kids are applying for FA, this was a bit depressing, but I appreciated knowing the reality.