MOSFET switch not stable
First of all, you have no fuses. The car battery can provide enough current to melt and burn your wires if you make a mistake, so that't dangerous.
Secondly, you can't use a N type FET as a high side switch. You need to either use a P type FET a high side switch, or move the N type FET to be a low side switch.
And third, the gate is floating so leakage currents can turn it on or off when no button is pushed.