We strongly recommend that users always run the latest version of Flash Player. Running an old version puts users at risk for malware infection. The vast majority of users are opted in to background update and are kept updated automatically, but there are users that for whatever reason have chosen to stick with an old, vulnerable version of Flash Player. While this is a bad idea, people are certainly free to do it. The upside is that the affected population that you're describing is pretty small.
The upgrade story will actually get better over time. The latest versions of both Google Chrome and Internet Explorer include Flash Player as a built-in component, distributed with the browser. Users are always updated with the latest version of Flash, and new updates are delivered inline via the Chrome updater or Windows Update respectively. Mozilla has spent a few years talking about how Flash needs to go away, while the other browser vendors have been investing in modernized plug-in interfaces and improvements in plug-in security, and unfortunately the prioritization is showing in the user experience. We're hopeful that Windows 10 will pick up the majority of the WinXP and Vista stragglers as well as many of the Win7 - Win8.1 users, which means that the whole Flash update problem for a very large population of people will be solved in IE once people upgrade.
It's hard to say what's at the root of the specific issue you're describing. With the major browsers on 4 or 6 week major update cycles (we publish monthly to ensure we can catch and address any new injections in the browsers that might come up in nightly/beta versions before they hit the general population, and to align with Microsoft patch tuesday), so you're talking something like 18 months worth of changes between the Firefox version you're running and the version of Flash that you're running. Everything at Mozilla gets done with bugs, and we'd definitely encourage you to file a bug on the user experience that you're seeing. They've been operating from a reactive place lately, and some of the changes that impact Flash definitely appear to be a little slapdash. It's ultimately good for them to hear from the developer community and not just me all the time, but feel free to add me to the CC: list in the bug.
We ultimately have no control over the longevity of a particular build of Flash -- there are way too many external variables -- and we're also investing a huge amount of effort in proactive security work (and it's been our top engineering priority for years). So each build usually includes large numbers of changes designed to either harden the player or address responsibly disclosed issues from researchers, the larger security and computer science communities and whatever the latest auditing and testing tools we can get our hands on are. The continuous stream of innovation and ingenuity in the offense space is impressive, but keeps us pretty busy. Browser-side changes can easily break older versions of the plug-in, and running an old version is just a bad idea.
So we've really streamlined the installation process to keep people on current versions, and Windows 10 will hopefully be a massive shift away from older, more vulnerable versions of Windows. The landscape is so different than it was even 4-5 years ago.. the days of running anything for years (or months) in your browser are long gone. The pace isn't going to slow down anytime soon, either.