Website Storage Settings panel not available in some browsers

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:

https://support.microsoft.com/en-us/help/4026529/windows-10-using-system-file-checker

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 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:

Use or fix Flash audio & video - Computer - Google Chrome Help

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.

The capabilities of marketers to track you using HTML5 and JavaScript have surpassed the capabilities exposed to marketers via Flash Player for years.  More importantly, if you live in the US, your elected officials chose to to let Internet Service Providers monitor your activity and sell that behavioral data to third parties for profit.  As we're thinking about generosity and gratitude this holiday season, you may want to send your elected officials a thank-you note for that little contribution to a better world...

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.