So why can’t someone at the DVLA apply common sense and enquire if you would like the address on your reg doc changing too? Just a little note, or even the form to do it with, in the envelope they return your licence in.
They will probably ask why they should be responsible for this, seeing as the car doesn’t have to be moving with you, may not even be registered to the person who is paying the tax etc. If they have to enquire every time someone changes their license address there would be a lot of phoning around going on.
FWIW neither of my cars are currently registered to my current address (though they are insured for this address) – everything still goes to my old address where I can pick up the mail anyway. I will get around to changing it one day.
A friend of mine forgot to tax his for several months (3 cars, slipped his mind), upon re-taxing (around 2009) he was fined £80 via a letter from the DVLA.