you CANNOT insist on your chosen repairer giving a quote. The owner of the damaged car is free to decide who will repair his car. Your friend can insist that HIS assessor views the damage. If the assessor disagrees with the repairs, ie if there is parts/damage to be repaired that is not accident related, then you can argue the toss. If the parts are needed, then the owner is entitled to compensation IRRESPECTIVE of whether he repairs or not.
Of course the taxi driver is entitled to loss of wages for the duration of the repairs and any other days before repairs that car was off the road. He must make a living - who will pay him if culpable driver does not?