Males and females human skeletons evolved differently for different load bearing weights in different areas. Sure, women carry fabric weight in giant gowns, corsets made of metal, and hoopskirts made of MORE metal. But keep in mind with the skirts and corsets 90% of that weight is rested on their hip bones and is then spread out over their legs. Yes, this will cause their spine a lot of pain and results in injury; one reason why we don't do it today. But those are also largely hollow objects and a full outfit could weigh easily 40 lbs, more if the unlucky chick is dressed in heavier fabrics like wool.
However, some armors could be as light as 55 lbs.
If you're in, say Europe, odds are you'll be in the Bronze age type of weaponry which is 90% metal armor / chain mail / very little fabric. Keep in mind that those armors required people to dress you into your armor usually and it wasn't an easy task for either party. Also note that the calvary knights ride on horses because it's harder for them to walk and they will move slower while on the ground.
Again, Europe has different areas of terrain. Think of Lord of the Ring movies if you aren't familiar with Europe topography - mountains, rocks, forest, and flat lands. most battles throughout land-waged history took place in certain areas of flat lands. The battle of Marathon took place in a very small physical area.
As you mentioned, this is important in battle. Part of the reason the North won over the South in the US Civil War was due to the invention and wide-spread use of a faster gun. Speed isn't everything, as the Battle of Marathon wisely proves, but it is a thing to keep in mind. And really what you mean here is "balance". Which covers the heel portion. If you have all of your weight on your toes and are required to thrust forward with a lance or sword and you know it's going to hit something, you need to be able to ground yourself for that force that will be defending. Now, if you're thrusting forward and won't be hitting anything besides something to propel you farther forward (pole jumping), then you need: Access to your entire foot and range of balance which means no heels. "I can run in heels!" isn't the same as "I can block a 250 lbs football linebacker running at me at top speed with a full-body metal shield and not slide 5 ft back without falling over". In heels, you'd be flat on your butt with the shield on top of you because your balance would be yanked off. Also, in general shoe-building from what I recall, heels are some of the worse designed portions and hasn't changed engingeering wise for decades. It puts your foot into a forward arch which shifts the bulk of your trunk weight forward from "standing over your feet evenly" to "standing in front of your feet evenly", which is why you "stand on your toes". I also vaguely recall something about heels, even chunky ones, have a breaking point to snap off due to their poor engineering, and that happening in battle will mean a sudden loss of balance and will result in the enemy killing you.
4 armor with holes
So actually, the real reason we wore tones of armor over our chests and heads are - surprise! - we have no exoskeleton. I won't go into all of the details, since that's better explained in a Spiderman theory of all things, but generally the idea is: Our bodies aren't designed to be hit against from the outside and the impact will cause our innards to slosh around and rupture, if not from the impact, from the impaling. We are literally very squishy.