I'm 100% sure despite what tescos say YOU can choose any repairer you want, but tesco would only be obliged to pay the lower amount/quote if thier approved man was cheaper.
The policy even if it had this in writing about using thier choice most likely would not stand up in court if needed, pretty sure it's illegal tying you into a certain person, so even if part of policy has no basis in law, ie doesn't exist in practice.
Take lots of piccies put things in writing, all tescos could insist on was that they were approved, though most insurers are happy to use anyone you are happy with.
At the end of the day they won't garage won't be paid until YOU are happy and sign to say so, so don't sign if a courtesy car has been supplied until you are happy, garages make a right sting from insurers on hire/courtesy cars so tesco would then take note.
tabetha