All of my tests and observations above were without the Noscripts add-on installed in Firefox, only a MVPS HOSTS file with
www.google-analytics.com blocked. Some folks commented that they have Noscript installed as well as a MVPS HOSTS file, and didn't experience the same problems with the avast website as I did.
I can now confirm the Noscript can change that behavior, depending on the settings. I'm definitely not a Noscript expert (only a newbie there), but I've found one setting which enables the avast website to work correctly. Maybe there are others:
Set Noscript to enable all scripts globally.
Disable the google-analytics.com script in Noscript at the avast website.
Then the avast website works fine.
That's counter intuitive. The reason is apparently that Noscript inserts some harmless surrogate scripts whenever a script is disabled, so the website is fooled into thinking the script has executed nevertheless. At least that's my theory.