Install Premiere Elements 2021 to a different drive then C:

The registry entries really shouldn't be a problem; for starters the registry entries will only be accessed by either Photoshop or Windows. If the drive isn't there and therefore Photoshop isn't there, then Windows will simply either complain and then silently ignore it and you can't use Photoshop anyway if the drive isn't there. So breaking links simply shouldn't be an issue.

Further, Photoshop is a big old beast and on an inexpensive 128GB SSD it's a massive chunk of the drive, and 2TB external HDDs are considerably cheaper than nTB internal SSDs. A lot of people, myself included, are trying to limit the use of the internal drive to be a system drive plus absolutely necessary programs. Even Microsoft have accepted that their reliance on increasingly larger internal drives is folly.

There are some programs that do have a legitimate reason for insisting on being on an internal drive (Ubuntu for example effectively forms part of the Windows Subsystem for Linux) but they are few and far between, and for some of use a graphics program - which is incidental to our day-to-day computer use - is not one of them.