No, He would NOT be liable for any towing fee (even though there wasn't one). It was the third party at fault, therefore their insurance company would pay any towing/assessment fees.
But yes you should have taken the car yourself to whichever repairer you wanted instead, then have gotten a quote and sent it to the company who was at fault (then they send it to their own Insurance company to assess)
The so called "Storage & Assessment" fees on the Crash repairs end is their own fee, regardless if you wanted them to fix it or not. BUT more importantly you should not have to pay those fees if they are repairing your car still, those fees SHOULD have been listed in the assessment report sent to the insurance company for the "Total Loss" estimate.