Flash update notices - make them stop! (Mac 10.11.6) 05/14/20

A screenshot of the update message that you're getting would be really interesting.  Google Chrome has a built-in version of Flash Player that's distributed and managed by Google via Chrome's automatic update mechanism.  You shouldn't ever need to download and install Flash Player in Chrome, unless something is messed up with the underlying system (the filesystem is corrupt or the disk is failing and Chrome can't write the update files, permissions on the filesystem somehow got mangled, etc). 

My guess is that you're getting this notification from a website you're visiting (either because Flash Player is disabled by default, or because a fake, malicious "notification" banner is getting inserted on the page), or there's a classic vulnerability in older commodity routers where they could be temporarily infected with malware (power cycling a vulnerable router should clear it until it's re-discovered and infected again), and the update notifications are inserted by the malware and point to fake downloads.  Updating the firware on your router would solve that long-term, but most people plug their router in and never think about it again.  Newer routers automatically update for this reason, but old, unpatched routers are still pervasive, which is why that issue has been around for several years.

Anyway, a screenshot in this instance would be super interesting.  

If you can't rememer having ever done it, power cycling your router and updating the firmware is a good move.  It's probably a good time to do that necessary, regular maintenance regardless.

If you're signed into your Google account on Chrome, it automatically saves your bookmarks and stuff.  You may want to just uninstall and reinstall Chrome.  If there are issues with the updater not working, that should solve it, and you'll get a shiny new Flash Player as part of the deal.  Chrome will sync your data back down from its servers, so it should largely be a transparent operation.

Details on enabling Flash Player in various browsers can be found here: 

https://helpx.adobe.com/flash-player.html

Would it be several minutes (or hours?) between the end of Step 2 and the beginning of Step 3 (which I assume is the password step).

Are you referring to the browser page step 2 & 3?  If so, this is the point where you go to the file system location that the downloaded file saved to and launch the installer from there.  This is the point where you will be prompted to enter the system admin password to install Flash Player.

To confirm if Flash Player PPAPI plugin has been manually installed, go to System Preferences > Flash Player > Updates tab. Please post a screenshot of the Updates tab

I have Chrome as my default browser and I don't even use Flash.

Google embeds Flash Player in Chrome and all updates to the embedded version are released by Google, not Adobe, as such, users are never prompted to update the embedded version.  There is no need to install it separately.  While Google does embed Flash Player in Chrome, it is disabled/blocked by default.  See https://support.google.com/chrome/answer/6258784 for assistance troubleshooting / enabling/unblocking Flash in Chrome.

I'm surprised Chat tried to help, they aren't supposed to help with free products and should have said so first time... If you HAVE updated Flash Player using the official Adobe download http://get.adobe.com/flash 

then you have fake updates. Millions of sites are trying to attack you. If a site pops up an update, never go to the download and never go to the site again. If you downloaded a fake update, your computer is probably infected and it's recommended to wipe it completely and restore from the last backup before you were infected. Flash in any case already includes Flash Player, no need to install or download it.