Most useful military headgear for a 18th century army

How come, European infantry (and most cavalry) didn’t get to wear any helmets? Helmets were all the rage from antiquity until the 1700s, and then again from the 1900s onward. Were they too expensive? Would they have been any good if people used them?
I’m talking real metal helmets here, not the fake leather imitations that the Austrians and Bavarians and revolutionary French wore. It doesn’t have to be anything fancy like a sallet or a great helm; I imagine that morions, or kettle helmets could do a better job than tricorns and shakos at protecting the head against sword blows and the weather.
I once read that some Bavarian general during the Napoleonic wars proposed to introduce Roman-style helmets for the infantry, but the idea was rejected.

If you were minister of war, what would you put on your foot soldiers’ heads?
You have an unlimited budget, but may only use pre-industrial materials.

The answer is guns. Guns happened. You’re not going to make a helmet that’ll protect you from a bullet to the head; at least not with 18th-19th century technology. So they just became a liability; heavy, uncomfortable, and not really useful.

In the 20th century, people started wearing helmets again because, while a helmet wouldn’t protect you from a bullet, it might deflect shrapnel, and shrapnel became a big problem in 20th century warfare. Additionally, medicine had improved, so that a head injury wasn’t always fatal, so long as the patient got quick treatment.

I think “foot” is the key word here. Soldiers had to schlep all their gear with them and any pre-industrial material thick enough to be useful is going to be heavy as hell. Which would you rather your soldiers carry, a helmet that serves no practical purpose, or a few extra cartridges of shot?

I was about to reply “but surely, a bearskin hat weighs as much as a hettle hat”, but then google happened. A modern British guardsman bearskin weighs 1.5 lbs. A medieval kettle hat weighs about 5 kilos. Ouch, that’s twice as much as a sword.

I dunno… the steel pot was also useful as a water basin, a cooking pot, a shovel, and had several other uses.

The helmets looked something like this, which seems like it would be highly impractical for pretty much everything you mentioned. Wasn’t all that great for protection either, but they certainly looked spiffy.

All right then, helmets are out, but what about the others? Shako, tricorne, bicorne, bearskin, mitre cap, czapka - which one gets the cake? Manufacturing cost is still irrelevant.

Right. This is why medieval body armor fell out of disuse. Guns could rip through chainmail and basic plate armor like putty, so the advantage that armor provided quickly turned into a liability.

Not so fast! Cuirassiers wore armour up until the 1870s, while the French kept theirs even until 1914. Obviously, they weren’t so useless. I have read in various sources that cuirassier breastplates were strong enough to deflect musket balls. Muskets have existed since the 1600s, and cannons are even older, but you can find pikemen dressed in plate and helmets up until the late 1600s.
I’m convinced that an armoured Napoleonic regiment would fare noticeably better against an unarmoured unit of equal strength. Sabre and bayonet cuts accounted for a substantial portion of injuries, and the piercing power of musket balls is often overrated.
I believe that the reasons for the demise of armour must be sought on the strategical or logistical level, and not on the tactical. My guess is that the armies had grown too large, and it was deemed more economical to suffer a few thousand more casualties than to dress them up in armour. From looking at it, it seems to me that you could forge two muskets out of a single breastplate.
Another reason could be that generals found it more useful that their armies could march quicker across the map, than to have fewer casualties on the battlefield.
And maybe the soldiers just hated wearing the stuff. I have read that the French Carabiniers complained greatly when Napoleon issued breastplates and helmets to them.