Re: My solution
Often, they are sites offering SaaS - 'cloud' applications. It's not always that they won't work in Firefox so much as the vendor insisting that they only support usage via Chrome and any bug that occurs will not be investigated unless it occurs using Chrome as they develop and test for Chrome.
I have clients that use Chrome, IE and Firefox all together, depending on the site. IE for a SaaS platform using Silverlight, Chrome for another SaaS platform (because that is what is supported) and Firefox for some government sites which require plugins that aren't usable on Chrome.
The fact that you, as an IT-savvy person doesn't want to use that site is irrelevant (no offence) to the rank-and-file employees who have to.
Sure, in a sysadmin's perfect world, we could dictact which browsers and add-ins can be used and everyone else in the company would seek our nod of consent if they wanted to use a site. But that's just not reality for the overwhelming majority of companies.
If you work somewhere where you can overrule a SaaS purchasing decision because you don't approve of the browser requirement then congratulations. But I have never worked anywhere like that.
The point is that browser compatibility is an annoyingly shifting landscape of vendors and versions and add-ins and compatibility and support can be fickle and equally annoying. If you've never been in the situation I described then, again, congratulations, but I have and I am sure I am not alone.