No. You choose who repairs your vehicle. Insurers rarely volunteer this, but steering you towards their own network is a commercial preference, not a legal requirement.
Approved repairers work to rates negotiated with the insurer, which keeps the insurer's costs down. That is a perfectly reasonable business arrangement — but the pressure it creates is towards the cheapest acceptable repair, not necessarily the best one for your vehicle.
On a non-fault claim the bill is being paid by the other driver's insurer, not yours. Your own insurer has even less standing to dictate where the work is done.
A manufacturer warranty covers defects in manufacture. Accident damage repaired to the correct standard, with the correct parts, does not void it.
A reputable repairer guarantees its own work. Ours is guaranteed for twelve months on parts and labour.
Occasionally there is an extra day while the engineer authorises the estimate. On a non-fault claim handled directly with the at-fault insurer, it is usually faster, not slower.
One call starts the recovery, the courtesy car and the claim at the same time.