I managed to fix the issue myself. It involved repeated downloads, because apparently downloading a whole page of store items at a time does not work well. Some selected items got skipped, some seemed to not be correctly downloaded. Anything that didn't have a proper thumbnail in the launcher didn't install. I downloaded those again, until I got all of them with a proper thumbnail. I now have all of my store content installed, unless I missed something. When my computer is repaired, I'll copy my entire downloads folder from there and compare to see what might be missing on my laptop.
Another thing I discovered, in case anyone else has issues with the store, is that if I do not allow the launcher to log in, I don't have the issue of downloads no longer starting unless I refresh the page, close and restart the launcher, or reboot my computer.