if you get new mission and helicopter is damaged - it's a bug.
if you get into damaged helicopter that landed on ally keysite - repairing process should be started, if not - another bug.
I know that sometimes landing pads are spread too widely so some of them placed outside of the keyside area (200x200 meters or something like that), so repairing not starting when you are on it. maybe both issues has same roots- AI land outside of the keysite and finish the mission, but helicopter not get repaired right after that (just a guess, you can take off and land on another pad close to the base center to check it out).