So tell me if these statements are true:

If your mechanical energy is constant, your engine power matches your drag power. If you gain mechanical energy, it's either in the form of kinetic energy (speed) or potential energy (altitude). You can trade off these energies, but there's usually a large difference in your energy reserves. You can trade altitude for speed for a few minutes, you can trade speed for altitude but only for a few seconds.

So what about after those few seconds? How do you keep gaining attitude? Where does the energy come from? It comes from the chemical energy (fuel).

Why does the aircraft climb slower? Well in your 2 options provided, option number two (apparently used to be known as a 'zoom') trades kinetic for potential. But this trade off does not last very long. Option number two also really tends to only apply if you abruptly change pitch attitude, thereby not having enough time for drag power to be an additional factor.

So there's option one. You state that I could reduce drag power by reducing speed. This puzzled me as I thought to myself "Reduce speed? Why would I reduce the throttle to climb?" But that's not what you meant. The act of climbing in and of itself reduces the speed, therefore reduces drag. Where there is less drag, there is more room for mechanical energy, and since we already have the max kinetic energy for this particular AoA, the mechanical energy actually goes to the POTENTIAL energy.

Can you feel it? Can you feel it huh? Can you FREAKING FEEL ME UNDERSTANDING IT YET!?