Public universities receive funding from a multitude of sources. Research is typically funded by grants, which may come from state or federal sources, but they can also come from alumni, fundraising, or charitable trusts.
Regardless, patents are a necessary part of invention. As others have pointed out, without a patent, what’s to stop some other entity from coming along and (for example) using your hard work to make themselves rich? I’d wager if it were your invention/discovery, you’d want protections too.
There are some entities out there that would easily abuse patents. But I find it hard to believe a public university would be one of them.