The Core

Asking anyone with opinions ...

What tools would you recommend for identifying accessibility problems around a website so they can be cleaned up?

The goal is to meet WCAG 2.0 Level A guidelines.

ANDI works as a bookmarklet and is meant to help meet Section 508 guidelines (the US govt accessibility standard, rather than WCAG).
https://www.section508.gov/test/web-software

I use these two contrast checkers a lot
https://webaim.org/resources/contrastchecker/
https://contrastchecker.com/

What standard are you trying to achieve? The Section 508 "testing methods" advice warns against automated testing. Ultimately, if accessibility is really important, you'll probably need to download a screen reader and navigate around your site.

Also, ideally, you would look at things in greyscale. There are some Chrome plugins that allow this

 - https://chrome.google.com/webstore/detail/grayscale-tool/odolflphhameojgliipcnahnipmogigo?hl=en
 - https://chrome.google.com/webstore/detail/grayscale-the-web-save-si/mblmpdpfppogibmoobibfannckeeleag?hl=en

An older A very young designer friend who is a  design-school graduate and has been a professional designer for over 40 years said that in those days, it was drilled into the head of every professional designer that they should always printed every design on a black and white printer before finalizing the design. The goal was only partly accessibility, but also to verify that their materials would reprint properly if the client wanted to printer a cheaper B&W version or that they would photocopy properly (unless they are top-secret documents, in which case they have red all over them because on old photocopies that rendered as black so that the subtle red overlay on the original would make copies unreadable).

OT: She recently went to get something printed at a local print shop and the guy was completely dismissive until they spoke for a while and he realized that she was one of those designers who learned to cut and paste back when that required scissors and glue, not CTRL-C and CTRL-V. Then he wanted to talk to her for hours about the lost arts and how hard he finds it to talk to "designers" today who never worked before Photoshop and Illustrator and know so little about inks and printers and so forth. Anyway, there's a lot of lost knowledge and a lot of it had to do with basic accessibility just because they didn't know if their designs might get reused as black and white printings.