I was just playing a game where a Gnome Fighter uses a battle axe one handed.
I realize this is allowed, Rules As Written, but is it too much? A 3.5' tall guy swinging a battle axe with one hand just doesn't work for me. In 1e and 2e, the same character would need two hands to wield the same weapon. In 3.5 and PF, the same character would have a smaller version of the same weapon (that did less damage). I have no idea what would happen in 4e, LOL.
Has anybody done away with this, or modified the rule?
Same me goes for longswords, warhammers, and basically any other one handed weapon that does 1d8 damage, except maybe a rapier, which is lighter (even if it doesn't have the Light property).

This is why when I designed the Caturday for Weird Races (5e) I stated that they use Dexterity for their claw attack and that it can be used as a bonus action (but if you do so you cannot add your Dex mod to the damage). An extra 1d4 claw swipe per round is not much, but it is something. We found out that Caturday are mook slayers. Turn one or two loose in a gang of kobolds and the kobolds die in droves (two attacks per round, one at 1d4+ Dex and one at 1d4 is almost guaranteed to kill one kobold and at least wound another, even better if they are wielding a weapon in one hand and not just their claws).
I do understand why the 5e designers made natural weapons for PC races weak. We have found out in playtesting various weird races that natural weapons can become a game breaker if not handled carefully. They can be used to turn any warrior into a two weapon combatant. A great axe wielding PC with a bite attack can inflict a lot of damage at once. Consider:
Str 16 (+3 mod), great axe 1d12+3 (avg dam 9) + bite attack 1d6 (avg damage 3) lays out 12 damage per round. Assuming you read the rules for two weapon fighting as including a bite attack (RAW says in the other hand, but a clear interpretation is that other it means two weapons and just assumes one is in your hand).
However, my design philosophy is that the PCs should be Big Damn Heroes and adding a claw or bite (or horns, whatever) as an option is awesome and compelling. Then again, I love the concept of bonus actions which seems contrary to the current design theory of WotC.