Yeah, a video capture would be really great for helping me escalate the issue. I did install the latest copy of ESET Smart Security as a free trial, and I was able to see the UI as expected.
You should be able to uninstall Flash Player, install an old copy from the Archives, and then you should get the notification to upgrade within a few hours.
Uninstall Flash Player | Windows
Archived Flash Player versions
(For 14.0.0.145, you'd unzip the file, then run 14_0_r0_145/flashplayer14_0r0_145_win.exe)
Then you should get an update notification either on first launch, or the next time Flash Player update service checks the current version.xml on adobe.com. I think the timing these days is that the update service will check after an hour of installation, and then every 24 hours after that.
I got the update request again day before yesterday, walked through it taking screen shoots, but this time it worked as supposed. I got the opt out question in the screen you also provided a screen shot of.
So now everything is fine.
What differs from last time is that the update file that is downloaded for me to execute is a new version:
install_flashplayer15x32au_mssd_aaa_aih.exe
The old was ...14x32au.... .exe
For me the problem is gone now, so thank you for looking in to it!