What causes me to stop gaining points in campaign missions?

There have been several reports online of people experiencing this issue. Apparently it is a known bug.

This post from 343 Industries claims they are working on some patches within the next week or so (as of 11/13) to fix various issues, including "A Content Update tomorrow that addresses some campaign fixes and scoring issues".

However, that "tomorrow" has come and gone. Can you make sure your game is fully up-to-date? If so, they may still be working on that patch.

EDIT: As of this update on the 15th, they have another patch slated for release on 11/19 which should fix "non-matchmaking bugs, and a variety of title-specific issues". So if the latest patch didn't fix the issue for you, that one might.

Halo: Reach, released in the December of 2019 for The Master Chief Collection, still behaves like you're describing. I'm almost certain that the score multiplayer is intended to eventually drop to 0x when you've exceeded the set "par time". You also don't gain points from killing the same enemies again (when returning to a checkpoint).