What you are missing out of your reasoning is the force of friction caused by the air, tyres and mechanics of the car. This is forces which slow you down and do act proportional to the speed. If you account for friction then in order to move at a constant speed you need to generate as much force as the friction which is a constant multiplied by the speed. If this is not the case then the car is going to accelerate depending on the difference in force between the friction and the force you are generating.
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