Why is there no lift component when calculating rate of climb?
Power and force are different things. Power is force times velocity.

The calculation is done in the reference frame of the air mass. In the reference frame of the aircraft none of the forces are doing work, and the engine power all goes to accelerating the air, but we don't have formulas for that.

The lift is, by definition, orthogonal to the flight path, so it does not do any work. That's why it does not enter the equations directly (it does indirectly due to reduction of induced drag at high flight path angles).

The “power needed to overcome drag” is needed for overcoming the aerodynamic drag. In climb the “aft” direction is tilted own, so part of gravity is added to drag and the “excess can be used for climbing” means the engine is able to produce the higher thrust needed to balance the sum of them.

Rate of climb is vertical velocity, i.e. velocity in the $z$ direction. But for that derivation to work, you need the velocity in the $x$ (horizontal) direction too.