Is this job legit? Or a scam?

It's a scam. Anything that requires access to any account of your's is a scam.

Total scam. They claim they will send you money that you are to distribute to "the employees." In reality, only your money is real and you are sending it back to the scammer under a different name.

The company is not on glassdoor, but it does show on the Better Business Bureau's site with no complaints. Says they've been around for over 10 years. I do question this: "to link a bank account that is made just for business with cash app and Zelle". If they're want to run cash through an account under your name, then stay away. If they want to pay you via Zelle and they only have access to send money to you (no way to take it and don't expect you to send it elsewhere yourself), then that's fine. Question them heavily on this to get proper clarification.

They want to use your personal bank account for payroll processing for companies? Scam is my bet.

On that, I would chat with someone at the University to make sure about that.

sure sounds scammy

Sounds like a scam to me.