No you don't, the government has extended the mot period from a certain date under the covid19 rules, they could possibly increase the extension period again if they wished.
The mot is a legal requirement set by the government not the insurance companies who have work to/within the law, classic cars, plant vehicles, tractors and other vehicle clases are exempt from mots by law.
I've yet to see a steam traction engine at a mot station/ hgv test center.
Good point about classic vehicles, I'm amazed that some owners of such vehicles have reported having problems with some insurance companies, breakdown/recovery services and even some cops over this aspect.
My motorcycle will be qualify for no mot and tax in three years time, I hope that things are crystal clear to all by then.