Nope, that makes no sense either (I don't agree with what I've been told as none of our bikes are plated), some guys I've ridden on track with don't have a license.
Unless signing in as a rider alters liability issues, then you could ride TB up to pit out.
The area outside the paddock is all public viewing and camping. By insisting on driver and vehicle street-licensing and a current sticker with current insurance, you effectively reduce and even remove the potential for spectators to bring their field cars and 4x4s, Razrs, full-on motocross bikes etc to "play" in those areas, and you also limit the number of 10-year olds being given driving lessons by their dads amidst all the other public there. Both of these used to be a big problem. Operators and spectators have been badly injured by such things, including a number of incidents of vehicles crashing into tents in the night. I think there was also one death in there from that many years where a dirt bike went through a tent, and I have video of a 4x4 crashing through a tent in the daytime.
That problem has gone away with the licensing requirements. Should a collision happen with what's left, insisting on street-licensed vehicle with current sticker and insurance reduces the track's liability exposure. They can point to the off-road vehicle ban as taking measures to stop potentially dangerous vehicle use in spectator areas, and enforcement as performing due diligence to maintain safety.
For any collisions and/or injuries that do still occur, the involved currently-licensed, stickered, and insured vehicle owner's insurance would end up being the first point of claim. Current sticker implies current insurance. Green plates are different. They are forever, and there is no government oversight of continuing insurance coverage on green-plated vehicles. That, and green-plated vehicles too easily fall into the "play" category and encourage play, and the race track spectator areas are not a place to play with off-road vehicles.
The paddock area is a different animal. The expectations of a safe environment there are tempered with the realities of that area being a working part of the race event. Crew and drivers are supposed to sign in as participants, and that places them under the event insurance whether they have driver's and vehicle licenses or not.
The track still has to exercise due diligence though, and that is why they impose restrictions on operator age restrictions and operation conduct restrictions in the paddock. An under-16 or older unlicensed operator isn't permitted to operate a motorized vehicle in the paddock unless they are a competitor doing so as part of their session. Recreational riding or driving in the paddock by them is a no. The paddock can already be a congested-enough place, and there really is no place for recreational riding or driving in a paddock, especially by under-age children.
Distances at Mosport are not that long. The track encourages walking, bicycles, and on bigger events they provide shuttles.