Insurance companies generally will allow this but at an out of network rate which then applies to your deductible. Of course out of network deductibles are generally large and the allowable is usually much less than the doctor bills.
Insurance companies use the model to keep people going to in network doctors who will accept at contracted rate. It keeps premiums lower and prevents some overbilling. A lot of doctors overbill and don't even realize it. For example, billing for a surgery and then also billing for an incision. The incision is obviously a necessary part of the surgery so it is included in the rate.
Insurance companies are not always the great evil everyone makes them out to be but they are obviously also a business and their purpose is to make money. Good insurance companies do it by trying to limit bad business practices, operate efficiently, and keep lower network rates. Bad ones over-deny services when they shouldn't. It's very difficult for a lay person to tell the difference when you are looking at it from the outside.