Aren't they usually franchises of a larger national corporation?

The HOA can hire a property management company, and the builder usually sets up some kind of contract, but after the builder is out of the way, the HOAs are always run by the home owners.

HOA board is elected by the homeowners. Laws are clear on this.

They might hire out property management duties to an outside firm

It's not about being lazy. It's that (like government voting) people try to confuse and mislead you with HOA voting, too. Like they bundle a bunch of things together and try to sell you the changes you like, while burying the ones that hurt you deep inside the text so you don't notice them. Or making the rules obscure enough that you don't realize their full implications until it's too late. Or, when they come to your door and ask for your vote to changes to the HOA rules, they (at first glance, quite graciously, with a smile!) request that you hold off on voting so they can discuss your concern and find a way to incorporate it into their changes... except it later turns out the package is all-or-nothing (changing it would invalidate existing votes!), and their real intention is to prevent you from voting against the initiative until the deadline passes, so that they could gather the minimum number of favorable votes more easily.