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.
I somewhat question the need, as well—at some of the contrast levels we're talking about, if your vision is that bad, you should really consider enabling a magnifier, color filter, or even a screen reader.
What becomes really problematic, however, are PDFs, which are supposed to be a quick way to offer printed documents digitally, without too much extra work. Making those work with screen readers can take a long time, and there have been some cases where clients decided to just remove them from their websites. I'm not sure what to do about that, but I'm concerned we're creating barriers to sharing information, with anybody.
The contrast issues sometimes make design a bit harder, but it's totally worth it I believe.