I’m glad to hear you’re considering questions like this! I too built plenty of inaccessible site over the years and regret not taking more time, even on my own, to make sure they were at least usable.
In the last couple of years the only time I’ve seen any conflict is when the developer claimed to build an accessible site or app and then did not. Clients might wish you told them about accessibility and expect you to know about it, but they also remember the tight budgets and timelines and most don’t have any unreasonable expectations. In just a few cases, I’ve seen client and agency split the cost of remediation, but it’s pretty rare. In fact, I’m betting in most cases your clients will appreciate a note alerting them to any issues and an offer to repair at a reasonable price. The a11y landscape changes often enough you can usually find something in the news to use a backdrop to your message. For example, there are many news articles related to the current waves of lawsuits hitting credit unions and small governments right now- just find a recent article, forward a link and say “I’d like approval for 10 hours to fix a few things”. For older sites, the increase in risk is a good reason to approach clients about building new site, especially if you have any non-responsive sites out there.
Otherwise, I’d recommending doing what you can as you have time and focus on any “blockers”. If you shared code across multiple sites, fix a component on one and then apply across others. For example if you use the same menu behaviors, write a patch and push it out. And then do the really easy stuff like adding a skip to main content link, lang attribute on the HTML element, and adding semantic regions.
Moving forward my advice to agencies and developers alike is to provide some level of education up front and having clients check a “no” box when they are unwilling to fund the small amount of extra effort to make sure everything works with assistive technology.