Typically, the greatest effort on the part of the horizontal stabilizer is during the landing flare. Here you’re flying very slowly, and so the stab needs to rely on airfoil shape, and it’s angle of attack in order to produce the required downforce (Note we’re talking horizontal stab AOA… not wing AOA). This downforce is used to raise the nose of the aircraft so that it can assume the two point landing attitude (nose high). If you put an inverted airfoil on the tail, you can create more down force at the horizontal stabilizer to do exactly this, raise the nose. The aircraft designer needs to make sure that the horizontal stabilizer doesn’t stall during the flare process. If the horizontal stab were to stall during flare, you’d suddenly lose almost all of the downforce at the tail, and the nose of the aircraft would violently drop. Not a desirable situation on landing for obvious reasons.

If I think back to my airplane design courses in university, it was always the flare condition at landing that we used to size the horizontal stabilizer. If it could handle the landing flare, then it was a fairly safe bet that it could handle any other flight condition.

Tom