On the other hand, if (like most people) you do not have vehicle rental insurance on your own auto insurance policy

Then you have no choice but to deal with the insurance company on the other side, and you will be subject to its policies, procedures, limitations and delays.  We cannot force the other side to pay (or to agree to pay) Enterprise (or other rental agency) "direct" for your rental vehicle (such that you would not have to spend your own money "up front" to rent a vehicle and then seek reimbursement).  But fortunately, most insurance companies will usually agree to do it anyway (even though they are not required to, and even though we have no legal way to force them to).  If the other side will not agree to pay Enterprise for the vehicle rental "direct", then you will have to "pay" for your rental vehicle by charging the vehicle rental on a credit card and by then submitting the rental vehicle credit card receipt to the insurance company on the other side, which (unless you were at fault) will reimburse you (usually before you actually pay the credit card company for the rental vehicle charge).  Either way (insurance company "direct pay" or charging the rental vehicle on a credit card and then obtaining reimbursement before you pay on the card), you usually do not actually spend any of your own money (out of your pocket) for a rental vehicle if you will obey the instructions below (unless you were at fault).