I think the easiest path into tech is through internships. Being a student at a well-known university can help get you in the door for good internships, but it doesn’t have to be an expensive private college. Some schools also have a “second bachelor’s” program in CS which might get you into the kind of internships you want.

I was in a similar situation as you. I did a 3-month “boot camp” and then worked at an unknown startup until FAANG recruiters started to email me, but obviously YMMV.