My apps won’t update

Yeah despite getting a blue dot, the apps aren't actually updating. This is easy to check in apps that have an about screen with a version number. I've actually watched what the device is doing if you do a update all. It will clean any apps that need it, then put them in "waiting". After a few seconds, it will give up and the app will lose the waiting text, but still be gray and finally get the blue dot. Meanwhile the App Store app may show a download progress circle, that never moves and reverts back to the update button. Tapping individual update buttons, appears to skip all that (or it's too fast to see).

This issue is affecting all devices on my account, even an old iPhone 5s and my Apple TV 4K.

For iPhones and iPads, the work around is to delete the app and download it from the store. If you don't want to lose data, you can offload the app and reinstall it (you might have to do that more than once if the app crashes on load after reinstalling). Both of these are a pain to do, especially if you have a lot of updates. Also there's no way to offload on the Mac or Apple TV.