I hesitate to post again having made a pigs ear once, and still a long way away from my Kermode, however
In a climb, during initiation and also once stable, if forces are resolved vertically and horizontally with respect to earth rather than the aircraft this should isolate the vertical forces responsible for the vertical component of the climb.
Amongst these will be a vertical component of the now inclined thrust, and acting in the opposite direction ,that is in the same direction as the weight ( or mass) of the aircraft, will be a vertical component of the now inclined drag force.
This will hopefully explain why although aerodynamic lift may be reduced in a climb, the combination of the vertical component of this and the vertical component of the thrust have to be greater than the S/L vertical force required, in order to balance the weight and the added downward component of the drag forces.