Summary...
No matter what I do the vast majority of the apps don't work.
Details...
I have been working on this issue since last Thursday, inclusive. After more than five days of trying, including help from an Apple Support call, I'm ready to just sit tight and see if Apple acknowledges the problem and what they do, but time is running out for High Sierra support.
My testing, while not exhaustive, has been comprehensive. Utilizing three different Macs of varying age, configuration, updates, software, users, and Apple IDs.
Everything folks have suggested I have tried, with one exception. That exception was the formating with APFS instead of HFS+, which still doesn't work. So, the issue...
...is not hardware, so forget about how your Mac is rigged.
...seems isolated to High Sierra and the App Store apps.
...does not present on other macOS or OS X versions.
...seems to affect about 80% of apps tested.
...presents as a crash report, or a message that it is broken or incomplete, or nothing happens at all.
...maybe related to the type of app it is (32/64-bit, App Nap support, etc)
...crash reports all indicate code sign error.
Best Scientific Wild *** Guesses...
Something changed recently in App Store in regards to how apps are delivered and installed. It seems to correspond to the introduction of macOS 11 Big Sur and the imminent drop of future support for High Sierra (possible end of the year?). I'm not surprised because macOS 11 is a dramatic departure from the previous iterations with processor and software support for 64-bit Intel and ARM. Apple Silicon may be related.
In an effort to have only apps appear in the Big Sur App Store that will run on that new OS there may be something unintentionally affecting the app's appearance and function via the High Sierra App Store. Pure speculation... Apple is using the digital signing of apps to manage and control what the App Store presents to users of Big Sur and did not adequately test the impact on High Sierra because speculation is that High Sierra support drops and the end of this month (November).
Additional Observations...
I noted that when scrolling through the 'Purchased' page of the High Sierra App Store that some of the apps don't link to their product page but do in the Big Sur App Store. Example: SketchBook Express was added to my account in 2011 and is still listed in my Purchased items in the High Sierra App Store and in the Big Sur App Store. However, the behaviour is different:
• Big Sur App Store - I CAN load the product detail page but CAN NOT install SketchBook Express (make sense, this app probably needs to be updated by dev);
• High Sierra App Store - I CAN NOT load the product detail page but CAN install SketchBook Express (doesn't make sense because it installs and runs fine).
Another example is TextWrangler from way back.
• Big Sur App Store - I CAN load the product detail page but CAN NOT install TextWrangler (make sense, DOES need to be updated by dev);
• High Sierra App Store - I CAN NOT load the product detail page but CAN install TextWrangler (doesn't make sense it installs but doesn't run).
I'm starting to see that it is not widespread. A handful of apps work, but the vast majority don't work anymore in High Sierra after the change last week. On Wednesday last week, one of my favourite apps, the Magnet app from CrowdCafé was updated and worked on my old installation of High Sierra. On Thursday after the complete rebuild of the computer with a clean install of High Sierra the Magnet app doesn't work along with dozens and dozens of App Store apps I use. So Sad :(
Current discussions I'm aware of are:
der-Josh
https://discussions.apple.com/thread/252032870
PeterPeterPeterPeter
https://discussions.apple.com/thread/252030900
GrenadeBait
https://discussions.apple.com/thread/252037712
(I wish I could change the title of mine because this has nothing to do with SSDs and HDDs.