Reddit could operate without subreddit moderators. The main reason mods exist is to remove abusive users and bots, both of witch could be handled by the vote system.

Reddit requires moderators in order for the business of Reddit to function.

no they dont. they literally have a system to democratically promote or suppress posts.

mods had unilateral control over their communities until very recently. short of doing anything illegal or breaking TOS, mods could ban whoever they wanted for any reason. what stopped this was the fact that communities would riot if mods were to ban random users they simply didn’t like. look at places like /r/latestagecapitalism, /r/blackpeopletwitter, /r/witchesvepatriarchy, or /r/conservative, they will all aggressively ban users or block users from posting if they do not go through verification or disagree with the group think. and the community loves it because they’re stuck in their echo chambers.

Reddit hires staff to do moderation

and if your neighbor hires a lawn care service, you should be paid?

Right, but you also can’t create a work agreement where one was explicitly denied. It’s like mowing your neighbors lawn then asking them to pay you, but they told you they wouldn’t pay you if you did it before you started. It’s the same with the 3rd party app devs too. While I think reddits actions are insane and detrimental to the health of the site, they are fully in their right to deny those devs access to their API and their site as a whole.

and reddit has it in their TOS that no one who is a mod is an employee of reddit.

they don’t have a contract, they’re screwed.