You have to pay tax on all benefits in kind, which is anything your employer pays for in addition to your salary. This includes the insurance premium, road tax, etc as well as the car itself.
You pay this to HMRC, not your employer, and the rules which govern how much tax to pay are completely outside your employer's control, so the fact that you pay tax on the benefit is just the law, and has nothing to do with any other issues relating to the benefits in question. Remember that they still pay the insurance premium, road tax, maintenance costs, etc, which amount to a heck of a lot more than the tax you pay on them.
Someone has to pay the excess in the event of a claim, and I don't think it's unreasonable that it should be the driver in question. It might require an amendment to the terms of your contract of employment, in which case they would need to act in accordance with that contract, but it is highly likely that there is a clause which allows them to vary the terms with so many days' notice.
If you don't like it, why not ask if you can opt out of the company car and take a cash payment in lieu of the car and related amounts?