The insurance company does represent the driver, within the confines of the agreement the insurance has with the insured. In short, if the insured wants the benefits of the representation, he or she will have to go along with what the policy says are the limits of the insurance company's obligations.
The insurance company would then tell the insured he/she is free to pay the injured person out of his/her pocket, but the insurance company will not pay for it since the policy only obligates the insurance company to pay if the insured was negligent. If the insurance company believes it can prove the insured was not negligent then it has no obligation to pay. Nothing stops the insured from paying anyway if he feels bad and morally obligated to pay, he will just have to do it with his own money, not the insurance company's cash. And that's where the rubber meets the road. If the insured truly feels bad enough to pay even though he may not be negligent he'll pay up with his own assets, to the extent he is able to do so. If he only feels obligated to pay if it's the insurance company that will pay it for him then he must not feel all that bad about it. After all, it's pretty easy to say the injured person should be paid when you aren't using your own money to pay it.
The lawyer provided by the insurance company does represent the defendant like any other lawyer would. But he/she also must tell the client that some decisions the client wants to make will either violate his obligations under the insurance contract or otherwise result in a circumstance where the insurance won't be obligated to pay for the damages. So the client is free to make the decisions he wants to make, but may have to pay himself to do it. The insurance company's obligations are well defined in the contract and the insurance company rarely will pay anything outside what the contract obligates it to pay.
Of course, a decision by the insurance company not to pay if the insured makes a certain choice no doubt will affect what the insured ultimately decides to do. That's no different than many other situations in life where you need to take into account the reactions of others when deciding what to do,