Speaking as someone that spent over a decade in the towing and recovery industry, you are far better off paying the service and then getting reimbursed through insurance.
I'm sure it's obvious that the credit card company doesn't actually dispatch a tow truck, they contract with a roadside service company (unlikely AAA, but I will use them for an example) who then contracts with the towing company.
There are a couple downsides to this, 1: roadsides don't typically pay very well so you will be the lowest priority (priority 1 is law enforcement, 2 is "cash" customers, 3 are accounts (repair shops and such), and at the very bottom is roadsides)
2, roadside dispatchers can either be terrible or decent, I've taken calls from them wanting me to change a tire over 100 miles away because they think we are close by
3 you may not be getting a decent operator with proper insurance coverage, someone that is working for bottom dollar simply has trouble maintaining everything necessary (this isn't a dig against someone starting out, it's just an unfortunate fact)
By paying up front for the service, you are able to talk to the towing company directly, they will know what to ask to find exactly where you are and as I mentioned before, it puts you at a higher priority.
I didn't mean to type a novel, be here we are 😁