I agree that following web standards gets you a lot in terms of accessibility and is a part of good programming and design.
There's a lot that automated checkers don't find. They can detect if an image lacks an alt attribute, they can't tell if an alt attribute's value is bad. Many are pretty good at at least flagging where there might be a color contrast problem for a human to double-check. They don't automatically find interactive components that aren't keyboard navigable or operable. They don't know when the DOM order of content doesn't make sense.
For me though, making sure keyboard navigation works properly early on speeds up the develop and test cycle. I also think that in many cases, perfect ends up the enemy of the good. It's a different conversation if a disabled person contacts someone about a site with a suggestion for improvement rather than it doesn't work at a all.
There's a lot that automated checkers don't find. They can detect if an image lacks an alt attribute, they can't tell if an alt attribute's value is bad. Many are pretty good at at least flagging where there might be a color contrast problem for a human to double-check. They don't automatically find interactive components that aren't keyboard navigable or operable. They don't know when the DOM order of content doesn't make sense.