Yes, just as carparks, etc., I agree a campsite is likely in any arising litigation to have been regarded as somewhere the rider might expect to meet other, public third parties. Not "private land with the land owner consent".
I most definitely would not want to be the one testing the legality, either as the rider or site owner, post a third party making a serious injury compensation claim against me.
E-bikes that conform to EAPC* rules can be used as a normal "push bike", so if they are allowed then without specific notice to the issue, so I expect the use of e-bikes would be deemed acceptable. Plus, these being legal we can get liability insurance for our e-bikes, as we can for our cycles.
Also, the confines of the campsite hardly afford much of the stated "shopping trips and exploring" as claimed assets for an E-scooter.
LINK*