My insurance is taking care of the claim but it can take up to 3-4 months before everything is finished.
You just stated that your insurance is taking care of everything AND you stated you chose NOT to take the supplemental insurance - how does this equate to a "problem with renting a car?".
I've had 2 car accidents with rental cars - never had a "problem", I called the rental agency, the took care of everything and contacted my credit card company to figure out the finances. I never got involved in the process except filling out the accident report.
Gerry