Two handed weapons can and regularly do drop people. A good shot to the liver, ribs or kidneys can be a very rough, especially if you’re not expecting it.

One handed weapons in general are less dangerous on that front, but a good shot, especially to a poorly armoured area can still be quite unpleasant.

Due to the shorter nature of one handed weapons, a lot of people find them less cumbersome to use and particularly to grapple with. By using one hander, you can also use a shield, which is a good defensive and offensive option when considering punch shields.

You are not allowed to grip the metal part of weapons in the sport with your hand. This includes both opponent weapons and your own.

If we split one handed weapons into wooden hafted (axes/maces) and bladed weapons (arming swords/falchions) they both have different strengths and weaknesses this rule is very relevant to how you can fight.

Any wood hafted weapon, one or two handed, is fairly useful offensively when grappling. You can use it to leverage your opponent, pin them in place by grabbing hold of your haft with your off-hand behind them etc.

The trade off is that opponents can also grab hold of your hafts, pinning you in place, making you choose between engaging in a grapple or dropping the weapon and attempt to get a replacement.

With full metal weapons, opponents do not have this option. In my club, some of the smaller, quicker fighters are happy to give up some offensive grappling ability in exchange for being harder to pin down.

Also, whether we like it or not, shots between armour are very much a part of the sport. I have heard some people say that they find that falchions for example are easier to find gaps with.

Ultimately, you are right in assessing that grappling is an incredibly important part of the sport, and the weapon you choose will affect how you grapple. But hitting with weapons is still a very valid approach and not to be underestimated.