"all vehicles must be taxed and insured and therefore cycles were not allowed"
While it's not a legal requirement, it is entirely possible for a cyclist to be insured. Arguably quite sensible in fact, and one of the main reasons for joining organisations like Cycling UK (née CTC). And cyclists generally pay all taxes owed; the same amount paid by other low-emission vehicles like electric cars. So do Costa allow all vehicles if the owner can show the correct documentation, or are they just lying about why they don't want to serve certain vehicles?