That isn’t a Comprehensive claim. It is collision claim with associated deductible (which you will get back if the at-fault party is found)
Correct. This is why I don't like the "file with the 3rd party insurance" advice that runs rampant here on TMC ... it's not usually to your advantage. At all.
Remember, your insurance company has lawyers. Lots of them. High paid ones. Ones whose job it is to extract money from others. And if you call your insurance company, they'll work for you. It's absolutely worth the deductible - which, if they're successful, they'll subrogate on your behalf and give back to you.
Insurance premiums are just money we pay for protection. At least get something for your money and turn the legal beagles loose.