Open Question: Should/Could Let my insurance company pay my medical bills?

I was hit from behind and incurred about 3000 in medical bills. The at fault insurance company wants to make a settlement that will just cover those bills. Should i have let my insurance company pay those bills. Seems like my insurance company was offering to pay them since i have medical coverage. But so i have only filed the claims through my work health insurance. I had to pay the deductibles out of pocket.