Should a tunnel be built?
For cars and similar road vehicles? NO
There already is one.
Do you think an additional tunnel meant for vehicles can be built? YES it could be built. Are you volunteering to pay for it.?
On a long tunnel like this there is too great a risk of incidents to affect traffic. A simple flat tire or running out of gas.
More serious stuff like a drunk driver or those of a criminal style wishing harm like a car bomb
The issues are not for the normal day to day operation 99 % of time . The tunnel has to be designed to also account for those rare occasional things that could happen.
A collision between two vehicles. Accidents happen between cars going the same direction.A tunnel like a bridge is easily blocked. YES of course you can have many lanes to reduce this. The cost of the extra lanes is many additional billions.
I assume you know the current tunnel is three tunnels interconnected.in many places.
One of these tunnels is a road tunnel.
You can get more vehicles through when they are bunched together on trains.You also avoid the issues of dealing with the issues of exhaust gases and supplying usable air for the drivers and passengers.
It would need lots of extra security stuff due to the current world terrorist stuff.
It could be built it would be very expensive and a security nightmare for those operating it.
Suggest you investigate Lærdal Tunnel in Norway similar to what you are seeking. It is under land and not as expensive or as deep under ground as a cross channel one would need to be.HINT you have to haul out and up all the stuff you remove to make the tunnel.
Having a place to do this near the middle really helps with costs. Doing so halfway across the channel requires an expensive but possible artificial island/ platform to hold the required equipment like air fans and air vents.The connection to the tunnel would not be very easy to do and prone to catastrophic failure. about 300 feet underwater.
You can see the already existing road tunnel.