I do not want heels in battle for women unless they are offered as an equally ridiculous option for both genders to traipse around the field breaking their ankes in the heat of battle. Equaly opportunity idiocy, FTW! Same goes for skirts. The battle skirt looking things in origins were fine but I do not want to see bits of fabric flying around to get stuck in something.
Actually, you know, since heels were once useful as equestrian footwear, and we now have horse riding...
Riding boots still have a heel to prevent the foot from sliding through the stirrups. Bad things can happen if your foot gets stuck in the stirrup when you, say, fall off and your foot is stuck so you get dragged by the horse. BUT, these are not heels in the sense that the OP is proposing (based on the images presented). They are square and no more than an inch in height. I cannot imagine mounting or dismounting a horse in actual high heels! Lol! Here's an example of riding boots:

I really don't know what kind of footwear mounted infantry wears. Most modern day riding boots aren't good for slogging through mud and what not as they are also slick. Awhile ago, I was leading my horse down a steep muddy hill as it was too dangerous to ride down. She was slipping and so was I. Let me tell you, having a 1200 lb animal sliding behind you is unnerving. Anyway, she started to slide into me and I accidently dropped the reins in my haste to get away. She turned around and took off up the hill into the forest. I freaked as dangling reins can be dangerous for a horse (they can get tangled and injure themselves) and tried to walk back up the muddy hill in my riding boots. I fell face first as I couldn't get any traction. I got up, tried again and faceplanted again. Luckily, I was able to coax her back to me. At any rate, riding boots aren't the best example of what people should be wearing in battle. Although, they are probably light years better than high heels!