What’s stopping me from just suing them for breach of contract rather than going through their appeals process if they illegally deny my claim?

Medicaid is a government provision that has no contract that underpins it. With something like United Healthcare, there is a bargained for exchange of mutual consideration (you give $ and UHC provides insurance subject to all the terms and conditions in the contract).

That is not something in Medicaid because Medicaid isn't getting anything in exchange for its health coverage. The government can put whatever restrictions on it that it wants. The benefit comes from the government having one more contribution to the ecconomic system. However, that is too attenuated of a benefit to say that there was a bargained-for exchange.