I've installed every version of calibre since late 2011 (0.8.?) on Windows 7, 10 and 11 and I have never had to use Administration mode. And I had no problems installing/running 8.7.0 here on Win10 or Win 11.
FTR: my gui.json has
"windows_register_default_programs": "2.54.0".
Try running in Safe Mode - if that works, then it's something external to calibre. If it doesn't work, try this:
With calibre not running, rename gui.json to gui.json.save, and then start calibre - direct from the .exe - it will create a fresh gui.json with default settings.
BR