Since we're talking about unaccelerated flight then there must be no excess off force in any direction so if one force substitutes for another, then the other must reduce by some fraction.
I think you'll find it easier to understand if you think of the forces as:
Weight: Acts vertically towards the centre of the Earth ie downwards
Lift: Acts at 90 deg to the flight path. The lift is produced by having the aircraft maintain an attitude that gives the wings an angle of attack, thus producing the lift. If the wing has appropriately shaped asymmetrical upper & lower surfaces then Lift can be produced at a reduced angle of attack. Reduce the angle of attack enough (even to a 'negative' angle of attack in the case of a 'normal' asymmetrically shaped aerofoil) and lift reduces to zero.
Drag: The cost of moving a body through the air and of producing lift. Combination of all forces acting rearwards along the flight path.
Thust: Acts forwards along the flight path.
In level flight - a horizontal flight path - the usual situation is:
Lift = Weight and they act at 90 deg to Thrust = Drag
So, the aircraft's weight is being opposed wholly by Lift. Any rearwards forces (Drag) is countered by Thrust.
Now think of a vertical flight path (aircraft pointing 'up'):
Weight: Still acts downwards. This happens to be rearwards.
Drag: Still acts rearwards along the flight path
Thrust: Still acts forward along the flight path. This is now 'up'.
Lift: Still by definition 'a force acting at 90 deg to the flight path'...BUT (!) if there was any amount of force acting at 90 deg to the flight path then it would cause the flight path to deviate from vertical. Looking from outside the aircraft there must be NO horizontal forces for the flight path to be vertical. To achieve this the fuselage is held in an attitude that ensures the wings are at an angle of attack in the relative airflow that results in zero lift.
So: Weight AND Drag both act downwards and at the same time rearwards along the (vertical) flight path. Thrust is the only force that is acting upwards so it must be opposing BOTH Weight & Drag.
In effect Thrust has completely taken over the job that Lift used to do, while still having to do its original anti-Drag function.
At any climb angle less than vertical then some amount of thrust is substituting for a portion of Lift.
Does this help?