There is a sentiment that accessibility isn't a checklist, meaning that if you're really trying to make a site accessible, you don't just get to check some things off a list and call it perfect. The list may be imperfect and worse, it takes the user out of the equation, so it is said.
Karl Groves once argued against this:
I’d argue that a well-documented process which includes checklist-based evaluations are better at ensuring that all users’ needs are met, not just some users.
I mention this because you might consider an automated accessibility testing tool another form of a checklist. They have rules built into them, and they test your site against that list of rules.
I'm pretty new to the idea of these things, so no expert here, but there appears to be quite a few options! Let's take a look at some of them.