(1) If you right-click the Add to Firefox button and Save Link As…, can the extension file be downloaded?
(2) If that downloaded, try dragging and dropping it into a Firefox tab. Switch to the Firefox window and after some delay, Firefox should show the extension installation drop-down from the left end of the address bar. Does it?
To detangle any downloading problem from from the verification issue, if you “unzip” the downloaded XPI file (you might need to change the extension from .xpi to .zip), does it contain the actual extension contents?
I think the time stamps are different because you haven’t yet extracted the files to a folder, which should cause them to have a current time stamp.
Could you test in a separate profile? It take a few minutes but doesn’t disturb your regular Firefox experience. This is just to check that the program files are not corrupted, and that we should suspect a setting/add-on/customization in your main profile.
New Profile Creation
Inside Firefox, type or paste about:profiles in the address bar and press Enter/Return to load it.
Take a quick glance at the page and make a mental note of which Profile has this notation: This is the profile in use and it cannot be deleted. That is your current default profile.
Click the “Create a New Profile” button, then click Next. Assign a name like Nov2021, ignore the option to relocate the profile folder, and click the Finish button.
Firefox will switch your default profile to the new one, so click the Set as Default Profile button for your regular one to avoid an unwanted surprise at your next startup.
Testing
Scroll down to Nov2021 and click its Launch profile in new browser button.
Firefox should open a new window that looks like a brand new, uncustomized installation. (Your existing Firefox window(s) should not be affected.) Please ignore any tabs enticing you to connect to a Sync account or to activate extensions found on your system to get a clean test.
Can you install any extensions in the new profile?
When you are done with the experiment, you can close the extra window without affecting your regular Firefox profile. (Nov2021 will remain available for future testing.)
Never the less, although I have attempted this in the past, I followed your instructions for a new profile, and the same result occurs when I attempted to install the downloaded XPI file (tab_reloader_page_auto_refresh-0.3.7-fx.xpi):