Well, yes & no: Remember, the front wheels rest at the front of the bed but, the front end protrudes beyond it over the A frame; the more you move the carried car forward the closer it gets to the towing vehicle, too close & you start restricting the turning ability.
Another thing to remember is, when carrying a vehicle, it should be balanced over the axles, ie:the front & rear wheels should be the same distance from the trailer axle.
One last thing, carrying a large vehicle means you need a larger vehicle to haul it...
Having said all that, I'm not sure plywood is a good idea for the floor of a car carrier...