I think that what you're seeing is that Flash Player is disabled by default in those browsers. A lot of content these days (particularly video on major sites) will seamlessly switch to HTML5, so it may look pretty much the same, except that Flash Player isn't in play. In the vast majority of cases, if you're not seeing "About Flash Player" at the bottom of the context menu, you're not looking at Flash content.
Modern versions of Opera are based on the Chrome codebase, and Opera 49 picked up the HTML5 by Default features from Chrome. In practice, this means that Flash Player is turned off by default in Chrome. You have to enable it on a per-page basis.
I don't have Opera-specific guidance, but it's basically the same browser as Chrome at this point:
That said, my question is really about what you think you're accomplishing by deleting Flash Player's local storage data. While you're absolutely welcome to manage what Flash Player does and doesn't store, I think it's important to point out that in the context of 2017, managing Flash Player's storage really doesn't accomplish much in practice.
There's some great academic research on this topic, but the bottom line is that the guidance that people used to give out about deleting your "Flash Cookies" is largely superstition at this point. Your browser's incognito/private browsing mode, coupled with a commercial VPN service is basically the only way that you get privacy from the eyes of the marketing industry today.
Anyway, to answer your direct question, you can also manage Flash Player's behavior by going to Settings/Control Panel > Flash Player > Storage will allow you to set storage preferences, and if you want to blow everything away, Advanced > Delete All will blow *everything* away. The one caveat here is Chrome, which embeds it's own Flash Player, and doesn't look at the centralized settings and storage that every other browser on the system uses. You do have to access Flash Player settings in Chrome from the context menu; however, in the vast majority of cases, Flash Player is off by default at this point.
I think it will be easier to number the points I'm replying to.
1) The Facebook games I'm playing use Flash. Once the games are loaded and I right click on them there's a menu which includes Global Settings and About Adobe Flash Player. If a game goes a bit wonky and you report this to the game developers the first thing they say is clear the Flash cache and they usually provide a link to the Storage Settings panel. Going to the Storage Settings panel and deleting the data for the game usually fixes the problem.
2) In my opening post I said "I found instructions for clearing the Flash caches in all browsers using the Control Panel. Flash (32 bit) is listed but, when I click on the link, nothing happens." I've clicked on other links in the Control Panel and they work so it's just the Flash link that doesn't.
3) I can play the Facebook games on Chrome, Opera, Internet Explorer and Edge. It's just the Storage Settings panel which doesn't show up for Opera and Edge.
4) I switched from Opera to Chrome which is my main browser now. I sometimes do a bit of practise using Edge, though. After the major Windows 10 update the other day I decided to check out the Storage Settings panel to make sure it was still available for Edge and discovered it wasn't. I don't use Explorer but I did check to see if it would play the Facebook games and if the Storage Settings panel was still available for it.
I'd like to know what to do in case the Storage Settings panel suddenly vanishes for Chrome as well. If one of my games has a major problem I'd like to tell the game developers that I've cleared the Flash cache and it made no difference.
The symptoms that you're describing (the Control Panel not launching) are really weird. This is not a thing that I've heard anyone else complaining about, and with a couple billion installations in the field, even problems affecting a minute percentage of the population result in hundreds or thousands of complaints.
First things first, if you haven't rebooted the computer since these symptoms started, that's a really good place to start, and I think there's a good chance that it will solve the issue of the control panel not opening.
If that doesn't help, normally I'd tell you to remove and reinstall Flash Player, but Flash Player is a built-in part of the operating system on Windows 10. I know for a fact that this isn't a widespread problem, so it comes down to *something* about the state of Flash Player on this particular system.
If you don't have Windows Update set to run automatically, you should probably run it manually to make sure that you're not missing a Flash Player update or something. There are definitely interdependencies between Flash Player, the OS and the browser, and they're intended to be a matched set. Having the latest of everything is definitely a good next step.
You could try running the system file checker in Windows to make sure that everything is in a good state:
After that, things get pretty murky. I don't have any other good ideas for a Win10 system, so it's going to be more about digging into logs and such to see if we can find any evidence of a problem preventing the control panel from launching. Again, this is super weird, and hints at a problem with something fundamental that we depend on to run normally.
I shut my computer down every night and have Windows Update set to run automatically.
It's getting late where I am so I'll run the system file checker tomorrow and report the results.
Just looked at the instructions for the system file checker. I'm not an advanced user and don't want to risk doing something wrong and messing Windows up. If I did I would have to take my computer to our local computer shop and they'll be very busy this side of Christmas.