Yes they should have informed you, if they were going to inform DVLA of a write off. You needed to know this, so that you could do whatever was necessary.
Suggest that you make a formal complaint in writing to the Insurers you had the claim with.
Category C Insurance Write Off – the vehicle is repairable but the parts and labour would exceed the value of the car. This is a tricky situation as there are plenty of amateur and professional mechanics who could use second hand parts to repair these vehicles at much lower prices than the list costs for parts and labour. Discuss this with your insurance company if you want to keep the car rather than have it written off.