I sort of feel that the rest of the answers are unnecessarily complex, given how simple the fundamentals here are:

###Momentum### Question: Is it necessary that L>m.g (or as you put it, an excess of lift) in order to climb?

Answer: No, at least not a sustained excess of lift. Newton's Laws state that an object in motion will remain in that state unless a force acts upon it. A force imbalance is required to set the aircraft into a climb, but once this has been achieved the forces can be balanced and the aircraft will continue to climb. As such, an excess of lift is not a condition required for an aircraft to sustain a climb.

###Energy### Question: Is it necessary that we add energy to the system (in the form of increasing our power output) in order to climb?

Answer: Yes, if energy is conserved then in order to gain altitude (and by extension gravitational potential energy), we must add energy. We could add no energy, not increase the power output of our engines, and simply pull up, increasing AoA but also drag, and we would climb for a short time as we trade kinetic energy for gravitational potential energy, however we would find that our aircraft quickly slows and we are required to dive to below our original altitude to return to steady level flight.

Hence a power excess is necessary for climb, but a sustained lift excess is not.

I sort of feel that the rest of the answers are unnecessarily complex, given how simple the fundamentals here are:

Momentum

Question: Is it necessary that L>m.g (or as you put it, an excess of lift) in order to climb?

Answer: No, at least not a sustained excess of lift. Newton's Laws state that an object in motion will remain in that state unless a force acts upon it. A force imbalance is required to set the aircraft into a climb, but once this has been achieved the forces can be balanced and the aircraft will continue to climb. As such, an excess of lift is not a condition required for an aircraft to sustain a climb.

Energy

Question: Is it necessary that we add energy to the system (in the form of increasing our power output) in order to climb?

Answer: Yes, if energy is conserved then in order to gain altitude (and by extension gravitational potential energy), we must add energy. We could add no energy, not increase the power output of our engines, and simply pull up, increasing AoA but also drag, and we would climb for a short time as we trade kinetic energy for gravitational potential energy, however we would find that our aircraft quickly slows and we are required to dive to below our original altitude to return to steady level flight.

Hence a power excess is necessary for climb, but a sustained lift excess is not.