Unresponsive at Fault Party
Generally, insurance companies will not accept liability for an accident until they have had a chance to talk with their insured. Sometimes, their insured is not very cooperative. If the at fault party does not have collision coverage to repair their vehicle, they have no personal incentive to contact their insured company timely. They might be hoping that if they ignore the issue it will go away. Even in a best case scenario where you will eventually have a police report placing the other driver as having been involved in the incident and at fault, it will take at minimum a month to obtain that report.
In the meantime, if you don’t go through your own policy, you will be accruing storage and rental fees* while waiting for the at fault party to eventually contact their insured. Going through your own insurance policy will mean you pay your deductible up front, but it will also ensure a much more smooth process.
*If you are unable to store your vehicle at your house and have to leave it at a body shop or a tow yard they will charge you money to keep your vehicle there. You will also be paying your rental bill out of pocket while waiting.