It's a flawed FBW system if it will counteract any upward momentum of the aircraft irrespective of the condition leading to the upward momentum.
FBW isn't counteracting upward momentum; it is responding to upward acceleration. And yes, FBW responds to upward acceleration regardless of condition because the accelerometers sense acceleration regardless of what caused that acceleration. That's exactly how FBW works.
This will invariably lead to lost of lift and a waste of energy used in counteracting the upward momentum that was originally generated by for example the runway surface (including that of a ski jump) or a upward air draft.
Slight movement in control surfaces isn't going to cause more drag than the wheels of the aircraft. We have seen from videos of J-10 and J-20 that the canards do indeed move in response to uneven ground, so your assumption that this doesn't happen is contradicted by video evidence. The FBW is only responding to what it thinks as aerodynamic disturbances.
In the former case, for example, it can just be as simple as a flip of the switch to tell the FBW that the aircraft is taking off and that it shouldn't counteract any useful upward momentum whether it's due the runway surface, an upward air draft at the surface of the runway or a ski jump.
Except it's not that simple, not to mention flawed. It's flawed because FBW has to be transparent to the pilot - that the pilot cannot notice the presence of FBW through the control eventhough he knows it exists. As I have pointed out already, adding a switch violates that transparency. Furthermore, the determination of how FBW behaves is a responsability of the designer, not the pilot. What do you not understand here that needs clarification?
It's not simple because when the aircraft is taking off, it is still moving in air. So, FBW not only have to work after the aircraft took off, but also have to work while the aircraft is still on the ground. Your idea of forcing the control surfaces to be fixed (ie. by switching off the FBW) while the aircraft is taking off means the pilot would never be able to take off because the elevators are fixed.
Finally, it's not momentum that FBW is responding to; it's acceleration. FBW is also not counteracting "upward air draft"; not to mention that there is no sensor to detect upward air draft. The FBW is correcting changes to aircraft velocities (u,v,w) and angular rates (p,q,r) caused by air disturbances.
During level flight, it should be quite easy for the FBW to determine whether the change in vertical momemtum is due to air pressure changes (as opposed to an upward air draft or a decrease in engine power), in which case it's probably more efficient for the FBW to try to continue maintaining a level flight by adjusting its control surfaces. In the case of an increase in air pressure, decreasing the lift of the wing comes with the benefit of reducing air drag. In the case of a decrease in air pressure, similar beneficial outcome can also be seen...
Maintaining altitude is the job of the autopilot, which works through the FBW system with FBW system having the ultimate authority. FBW responds to acceleration, while air pressure and airspeed are used in determining how forceful the response should be.
...but the same cannot be said in other cases, in particular during take off as explained earlier.
Acceleration is acceleration, regardless of whether it is generated in the air or in the ground. The FBW responds to acceleration, and yes, it occurs on the ground while the aircraft is taking off as well. We have seen J-10 and J-20 canards move in response to uneven ground. Your assumption of how FBW works is purely based on speculation. Why don't you read up on it instead of arguing on a subject which you don't know?
