It is unfortunate that we get a common usage of “power” that leads us astray here. In many ways power is actually the least important.
Power is a measure of work done over time. So it is a useful metric to understand what is happening, but it isn’t the thing that does the work.
If you want to move something, you apply a force. Pushing on something is applying a force. If you want your car to move, you push it. When it moves, you are transferring energy into the car, and the rate at which the energy is transferred is the power. (If the darned thing won’t budge, rather counter intuitively, no work was done. So no power.)
So when you say “power is applied” just swap out power for force. You can’t apply power to anything. You apply force. Power is how you measure the rate of energy transfer.
Think of the entire car’s system in the same way you do a bicycle. The pedals are the pistons, the crank is - well - the crank. You have a chain transmission with gears, same as the gears in a car, and eventually you try to rotate the driving wheel. Expanding hot gas pushes down pistons, your leg pushes down on a pedal. Nothing else is much different. The mass of the pistons is like the mass of the pedals. They are pretty inconsequential in this view of things. They need to be strong enough, but they are coupled to the rest of the vehicle, so the expanding gas doesn’t see the mass of the piston, it sees the mass of the vehicle. Same as your foot doesn’t see the mass of the pedal as you push down. It sees the mass of you and the bike. (Mass of the piston does matter, but where it really matters in on top of the upstroke, when it tries very hard to pull itself off the end of the coupling rod. Your bike doesn’t have this problem. You can’t pedal fast enough for this to matter.)
As above. Torque and power are related, but one isn’t a subset of the other. Torque is the force that is pushing things around. Power is a measure of the rate of work done. You can be in a situation where you are applying a lot of force (starting the whole rig moving) and thus need a lot of torque, but the actual power delivered is very low, since you are moving quite slowly. It swaps at highway speeds. You only need enough force to balance the air drag, friction, and transmission, but you are moving quite fast. Thus the power delivered may be quite high. Eventually you can’t go any faster. (As a rough approximation, car designers used to work out the balance of these losses and choose the transmission ratios, so that in top gear, your engine is delivering maximum power when the car is travelling at its top speed. This ends up with poor fuel economy, so they add overdrive ratios, that get you good fuel economy at highway speeds, but are too tall to allow the car to reach its maximum speed. So your maximum speed is achieved in say 5th in a 7 speed box.) The balance of peak torque and peak power in an ICE is tweaked to give good driveability. Diesels provide a better starting point for getting the balance right for moving lots of mass. Gas engines are more flexible, and provide a good balance for lighter, zippier, things.
You can pick a lower gear, and this is exactly what you do. But your gas engine may be revving higher than you like, and generally not as happy as a diesel. (It will be wearing much faster for a start.) It depends upon how it is designed and tuned. Neither engine type is ideal for all circumstances, but generally diesels are good at low rev torque, and are happier for this task.
Getting moving this is hard on the transmission, especially the clutch. An ICE can’t do anything if it is stopped, so you need to bridge the gap. That means a sliding clutch or a fluid coupling (aka torque converter in an auto transmission.) Either way there is a mechanism of allowing the engine to run whilst apply torque to a stationary vehicle. Getting past the use of this as fast as possible reduces the wear. Slipping the clutch trying to get a large rig moving will work, and kill the clutch real fast if you keep it up. Or kill it first time you try if you get it wrong. Automatics are dissipating heat as well, and they is why you often need a transmission cooler for towing. All the energy from the engine not going into moving the rig is going into heating the fluid. It can’t keep heating up forever. So the answer is to get the transmission solidly engaged asap. That means having an engine that is able to deliver the needed torque to accelerate you at as low a rev rate as possible. That favours a diesel. Or electric. Cybertruck maybe? 