I’ve been a member of CTC, LCC and BC.
All have similar and encompassing 3rd party insurance.
CTC – best mag which is quite good, good discounts throughout main London stores and on some organised rides.
BC – no mag, Halfords discount (via voucher download) but preferential application for BC events, esp track at the Olympic velodrome.
LCC – Mag which is 95% about cycling campaigns, etc … same discounts as CTC.
And you’re right to get insured riding in London IMO.