Yes some companies gatekeep based on college, but A there aren't many of them, and B they often only do that for fresh grads. It's not worth worrying about those places when they will happily hire you in 3-5 years anyway, and you likely are going to leave again in another 3-5 years.

University of Phoenix is a step up from ITT tech level scams. It's not really the same league as as the 13/15 state universities you haven't heard of in my state (Michigan), and probably 10 or so of the private schools.

You also generally can't get a bachelor's degree at a community college, so they really aren't at the same level either. They aren't a bad option to start with though.