Updated: December 29, 2012
Google prides itself on offering a seamless experience with their browser, Chrome. In other words, you are not supposed to see or feel any changes in any way. Your usage model should not be impacted. Only this unwritten rules break sometimes.
The issue you are facing is as follows: Since you received a new silent upgrade in the background that jumped your browser version from say 20 to 21 or so, you are no longer able to use Flash properly. You receive yellow prompts at the top of web pages asking you whether you want to run Flash this one time or always. And no matter what you click, the Flash plugin cannot be loaded. You start fiddling with settings, options, allowing and disallowing plugins, but this does not make a difference, either. Let's fix it.
So this is what happens, like the screenshot above shows. You hit either one of the two options, and you are inconvenienced. The placeholder where you are supposed to be seeing the Flash animation reads: Couldn't load plug-in. Awesome. Not.
Attempt to fix
You open the settings menu and try to make manual adjustments. The first thing you do is expand the privacy options, and then under the Plug-ins sub-category make sure they are allowed to run automatically. You also check the exceptions, but this does not work either.
Next, you head over to chrome://plugins. You mark the Adobe Flash Player as Always allowed. However, neither this change makes any real difference.
What you need to do, under chrome://plugins is to expand details and then disable the internal plugin. As it turns out, Google Chrome can use two different versions of Flash, its own version, which comes bundled with the browser, and the native system plugin, which you have probably installed on your own.
For example, below, we see that Version 11.5 of Flash is in fact Pepper Flash, and it is installed inside the Chrome application folder. On the other hand, Version 11.4 of Flash can be found as a standalone DLL inside the Macromedia folder in your system path. If you disable the internal plugin and only leave the external one active, you will regain the Flash functionality without any stupid questions. Please make sure you disable only the internal one, not both. So do not checkmark the Disable button at the bottom of the expanded menu, but the separate one for the internal plugin only. I hope this is clear.
If you look at the setup, it's quite confusing. First, you have two files for the Adobe Flash Player. The version is marked as 11.4, but inside the detailed list, you obviously see two different versions that do not really match. You have the confusing information like PPAPI and NPAPI, which only geeks understand. You can disable each one separately, but you do not know what the stability, functionality and security implications might be. And you can also accidentally Disable the entire plugin functionality, too. What does Always allowed mean if this option is already marked inside the Settings menu, as Run Automatically? Someone really botched this one, if you ask me.
This short tutorial teaches you a few things. How not to panic, how to approach the problem resolution systematically, how to look through Chrome's somewhat hidden setting and privacy options and manage plugins and their exclusions, how to enable/disable plugins and different versions of these same plugins, and finally gain back the desired Flash playback capability.
Google Chrome might be elegant and nearly transparent, but when it comes to problems, it is no dandier, easier or more configurable than any of its counterparts. In fact, the forced simplicity and the background update process that duplicates the native system behavior do somewhat complicate things. Well, I hope I helped you there, and now you have your Flash back. Mission accomplished.