Well I can't speak to minor damage.. However we totaled a deer in Wyoming a couple of years ago on our summer vacation.
$15K Damage to the mini van. Called the CC company about 30 minutes after it happened got all the instructions from them. Followed them to the letter. Rental company (Alamo) dropped off a replacement Van the next morning and continued on our journey.
When we returned it (the working Van) I expected them to need us to sign paperwork or something. Guy said nope glad you were all alright we will be in touch.
The sent me a bill, CC company took it from there. They went back and forth for a few months (everyone told me on both sides it would take a few months). In the end got letters from both noting claim paid and cleared.
Honestly it was a very straightforward process. Just make sure you contact the CC company and document, document, document...