You don't just sign a contract with the HOA because most HOAs are not voluntary. There are some voluntary HOAs because they provide community amenities and you get access to those amenities when you join the HOA and possibly agree to some other restrictive covenants as well. Most HOAs you never even have to sign a contract it is written into the deed of your property that you are required to be a part of the HOA. In order to own the property you automatically are a member of the HOA and subject to all of it regulations.

What does a government do? They make rules, enforce rules, collect money, organize some services, etc. All of this is done by the HOA too. You vote on leadership, just like citizens of a state. In essence, the HOA is an additional level of government.