You could argue that as the rail tunnel is not at anything like full capacity it would be wrong to add another cross Channel link. Clearly the owners of the rail tunnel would not welcome a new competitor, and were not expecting one under the terms of their concession.
You could also argue that maybe a road link would be more popular and better used than the rail one. Whilst a new road would doubtless do considerable damage to the business model of the rail tunnel by taking substantial traffic away from it, it might also generate some additional traffic of its own. If more French people came to the UK as tourists that would be a bonus for the UK economy. If more UK people went to the continent to shops and holiday it would be bad for the UK balance of payments, but might be welcomed by those taking advantage of cheaper and easier travel.
It is difficult to see the Channel tunnel keeping much of its shuttle business carrying lorries, if they were able to carry on driving to get to the UK. That is the mainstay of the tunnel’s freight business, which would be badly affected. Passenger traffic is more difficult to gauge, but again there could a lot of people who would like to go by their own car instead of taking the train and then hiring a car or using taxis when they get to the cities on the continent served by the trains.
My advice to the government would be not to commit any public money to a road crossing. They should also check the legal position carefully over the concession to the current Channel tunnel operators and owners. There are many other road projects we need in the UK that should take priority for limited sums of public capital. If the private sector wishes to design, build, operate and finance a road link then of course the government should be willing with the French authorities to examine the scheme to see if it deserved approval and support from the government as regulator. It would need to be built with artificial islands to avoid ship collisions with its piers, and would need to leave plenty of rooms for deepwater shipping lanes in what is a very busy piece of water.
The government would need to consider the security and borders consequences of a road link, given the difficulties the rail link helps create in Calais today. It sounds as if from additional briefing there are no current plans for such a scheme.