From your link re Michigan no-fault, with my [comments in brackets] and my bolding that applies to OP scenario re repairs to C’s car:
If you have an auto accident, no-fault insurance
pays for your medical expenses [Personal Injury], wage loss
benefits [Personal Injury], replacement services [Property Damage], and the damage
you do to other people’s property [Property Damage]. It does not
matter who caused the accident.
Your basic no-fault insurance does NOT pay for
repairs to your car. [Property Damage]
re: OP using basic/minimum required insurance coverage:
No-Fault = C repairs would be paid by B’s insurance, and quicker since there’s no need to determine fault.
At-Fault (Insurance only has to pay when insurance determines who is at fault) = C repairs would be paid by either A or B, or both, depending on who was at fault. I can see an argument that it was A’s fault (B claims A was only trying to avoid being hit by A, and that A caused B to hit C). So B’s insurance would not pay until forced to, and may only pay a partial amount. A is either long gone or insurance would not pay/partial pay for same reason since there is an argument A is not at fault for C’s damage.
Outside of minimum coverage, hopefully C has good insurance that covers damage to his own vehicle. In that case, C insurance would pay, and quickest.
Fault still matters of course (see post 30 re insurance subrogation), and the above just speaks to who would pay C and not if the insurance company would later get reimbursed or not.