I have found this is a widespread problem, widely reported symptoms on different forums are remarkably similar. It is mainly but not only Microsoft's own apps which are affected. The reason why Power Shell cannot re-install / re-register apps, in every case is stated as "Package could not be registered because there was a merge failure." Meaning?
Many more apps than I first realised have disappeared, because some are ones I rarely use, such as Calculator, Maps
I too have the weird Other list at the bottom of All Apps showing (in my case) nine none-functioning apps, all of which are named strangely like this "@Microsoft.People_xyz" ---- xyz is a very long mainly numeric code. In a few cases these names appear on none-functioning tiles.
None-functioning Groove Music is listed like that but is referred to as 'Zune Music'!
Advice to uninstall / reinstall apps misses the point, because you cannot find the app to uninstall it and you cannot find Store to re-install it. I have tried unsuccessfully to download / re- install some apps like Films and TV from the online Store (which I can launch). What I get is a message saying 'you will need a new app to open this Microsoft Store' with no explanation and no way to do it. What can this 'new app be'?
I have used SFC / SCANNOW and DISM which report 'there is no problem'. Who are they kidding? I have used FixWin 10 which uses Power Shell and various Troubleshooters. I used four restore points, which did not work.
It really is a mess. As many people are suffering (and more to come?) I do hope Microsoft will issue a fix, soon. One suggestion is to re-install Windows 10 with the Media Creation Tool. I paid a technician (quite a lot) to install it in the first place. It is not an attractive proposition.