The CBT lasting two years is the thing that seems strange to me.
You undertake the training, get given the certificate, and are therefore free to ride a 125 around without any further training or a license, at any time of day/night/year - but only for 2 years, after which you need to do the same again if you aren't looking for any further progression.
I can't see why you couldn't have a license limiting you to a 125 - without having to go through the full A1 and A2 courses which just cover all the aspects you have likely been doing on a 125 over the past two years?
My cousin has been riding a 125 for all of her biking life (probably 15years or so), no intention of getting anything bigger, 125 works perfectly for her commute, but still had to undertake the full test procedure just to stop re-doing CBTs, and to ditch the L's.
As for learning in the winter, I'm doing it, it's lovely