How do project managers navigate the challenges of prioritizing quality objectives amidst project constraints like time, cost, and resources?

The answer may be in your question. Many people, in the IT industry at least, seem to focus on quality control - checking after the fact to find defects. After all, testing is a big thing in IT. But quality assurance is arguably more important that quality control, to make sure things done right the first time, rather catching it after the fact.

Creating an atmostphere of quality assurance by taking the time to set up in advance and predict where processes might go awry is, I think, key.

You still need testing, of course, but doing it right the first time is far superior to catching errors later.

The biggest issue most PMs have with prioritizing quality items is not being able to evaluate their relative importance. Theory of Constraints tells us that we need to identify and focus on the constraint that has the most impact to the desired goal. Often, those constraints are not at all obvious.

I have seen many PMs take on a project where there are a very high number of potential problem areas, and not know how to figure out where they really need to focus. This turns into "analysis paralysis" where they become overwhelmed with which way to go among all the choices and so they go nowhere.

I had some breakthrough experiences earlier in my career where I was able to analyze a problem by modeling it and then start tweaking values to see how much they affected the outcome. Some seemingly very important variables actually had little influence while some that seemed trivial dominated the overall results. From that experience I have developed some of my own heuristics like "A small angle over a long distance becomes a big problem."

It works the same in business systems as it does mechanical ones. Between a sub-team with a 100 hour deliverable and another with a 10,000 hour deliverable, focus on the one where a small % error can be a big issue over time.