While you are correct to say BS7671 is not statutory, it may be used against you if something bad happens and it has not been followed, so it sort of is in all but name.
Back to your caravans parks and whatnot, the owners/landlords do have a statutory duty care to ensure they are safe for use, and must take reasonable precautions to ensure that is the case, having the EICR at the recommended intervals done is one way of proving they have taken reasonable steps.
While it may not be against the law per se to omit these EICRs, they may find that their insurance is void if they don't.
It need not be necessarily an EICR, for instance bigger firms/organisation may have their own maintenance and record keeping personnel which is another way to prove reasonable steps have been taken