As a fellow Brit I’ve always been puzzled by this too.
The DVLA website and online processes are very good. Which is rare to say about a government IT system. I don’t think I’ve ever had any problems for a decade (or two?) dealing with everything driving-related fully online.
I expect since we just have one single system, and a smaller number of possible workflows/situations to cover, it’s easier to create a central online system. In the US, with 50 different variations perhaps it’s a harder problem to solve.