I ran some numbers on my calculator. 80 skill 39 armor vs Dwarf Guard - a classic grind mob.

Scenario	Damage dealt	Damage taken	Damage / taken
GS Offensive	57.79	13.58	4.26
GS Balanced	40.04	7.43	5.39
GS Defensive	10.97	2.59	4.24
F Axe Offensive	38.18	6.86	5.57
F Axe Balanced	25.62	2.89	8.87
F Axe Defensive	5.84	0.95	6.15

From this Fire Axe is about 3% less costs but 5% slower. ( Giant Sword on Balanced vs Fire Axe Offensive). This may be made worse for Axe when considering Axe’s worse training efficiency (mentioned by Dratini).

Fire Axe on Balanced takes about 60% the damage of GS on balance - a significant improvement although with 60% of the kill speed.

So I agree with your point that you can save a lot of  UH with Fire Axe on balance, if you are happy with that slower kill speed.