Before doing screen reader testing on complex web components, what I see as some kind of lack box testing where you test your whole screen reader + browser stack, it is useful to have a look at what the browser passes to a screen reader. Especially Firefox has a very nice accessibility tree panel in the devtools these days. In my experience, the more visual tree that is shown there is also easier/faster to read for users that are not blind and are not that quick when using screen readers.

Also, keep in mind that something that technically works correctly with screen readers is just the beginning. User testing might reveal lots of issues you wouldn't think of yourself. And yes, I know that resources are usually limited and there is not much room for user testing, especially testing with screen reader users and other groups that have some kind of disability. I recently worked as the accessibility lead of a mobile COVID exposure notification app that had a very simple UI and a hard accessibility requirement. We had the luxury to do extensive user testing and even in this simple interface we found lots of small changes that improved the experience for screen reader users.