I grew up just outside the great park and learnt to use gears on the road across Smiths Lawn. We used to enter by Blackness Gate and exit by Cheapside Gate for a 7 mile loop.
As a family group we never experienced any difficulties, though the area of the village, the deer park and the long walk was not where we went. It may be that entering from bishops gate placed riders into a more contentious area with more walkers etc.
It should also be noted that the park has a 20 mph speed limit, which as children we were never threatening, but I suspect I would now have to actively observe.
None of this is a good justification to limit cycling in the park, and I wonder what the impact on NCN route 4 through the park is.