There is work being done, but not by lift.
As long as there is thrust and the airplane moves, there is work being done.
An airplane with engines delivering 5,000 Kg of thrust that flies 100 km from A to B has made a work of 500,000,000 jules between those points.
In a steady climb, we have first to establish a few assumptions. In a constant EAS climb, for instance, TAS increases. So there is an acceleration, as well. The energy to increase both potential energy and kinetic energy comes from the fuel, too, along with the energy used to compensate drag and other energy losses.
If we switch to a constant mach climb, below tropopause (OAT decreasing with altitude) TAS will decrease, as well as EAS. So some kinetic energy will be traded for potential energy, knot by knot, and is felt inmediately in the VSI. Reducing EAS should help because of the reduced Drag. Above the tropopause, TAS will become constant again. Only EAS decreasing will help in the climb. On top of that, we actually climb at constant IAS, which at high mach numbers over reads with respect to EAS, which complicates calculations...
Depending on the particular problem is easier to use energy or forces. You can look at a steady climb from many points of view. From the point of view of energy, if the state of energy of the airplane increases, energy must come from outside of the system (fuel in the tanks is considered out, for this matter).
Similarly we can explain Lift using energy conservation (Bernoulli equation) or using forces (newton laws). Or such a thing as the koanda effect, which I will never understand.
Energy is like an accountability book. If everything is OK, the number in the left page must be the same as in the right page.
Forces explain in more detail what is going on. To me, Newton's point of view to explain Lift is better than Bernouill's.