So I guess the answer is that there are trails on and around Mt. Ascutney to ride, just not within the State Park. Then again, the trails I have hiked within the State Park seemed mostly/totally unrideable, so no big loss- you have alternatives nearby.
For one reason only, you don't want to try to ride the trails in the State Park- hikers. The trails are full of slow old folks, families with fat kids and stupid dogs, etc. Not much fun as far as I can imagine.
If you love pain, they have a hillclimb race up the paved road every summer.