The most important thing to remember about military uniforms throughout history is that economic concerns were almost always more important than practical ones. Availability and ease of manufacture colluded with fashion to create the look of an 18th/19th century army.

From a practical standpoint, however, a shako was (in my opinion) an improvement over the cocked hat of the 18th century. It had a brim, which did a far better job of protecting a soldier from the sun than the brimless cocked hat, and the shako combined with a cockade created an impressive soldierly silhouette on the battlefield.

As far as protection, a shako had very little to offer. The British, for the most part, used a pressed felt shako that was very thin, and a good sword cut would have virtually no trouble passing through it, to say nothing of a bullet or shrapnel from artillery. The shako itself was so fragile that the felt shakoes were issued along with a canvas cover that would be placed over top of it in inclement weather. American forces in 1812 used a very thin leather shako which didn't need addition protection from rain, but would provide similar (read: negligible) protection against meaningful attacks.

Which is not to say that a shako was a better hat than, say, a straw hat, or even a kepi, which as you mention became relatively standard by the mid 19th century. Kepis, too, had problems. They were hot, and just like the shako, only offered a small brim to protect from the sun and virtually nothing else. Kepis were even less useful when you take a look at some of the "dress" versions of the kepi that included stiff ribbing to make the kepi stand up rigid and created a hat that was simply a cloth version of the earlier shako. Removing the ribbing allowed the kepi to droop, giving the somewhat prototypical look of the American Civil War soldier (you'll see these limp versions of the kepi in most Civil War movies, somewhere or another).

Most of the field troops in the US Army by the 1870s and 80s wore "campaign" hats in the field, which were soft felt wide-brimmed hats that resemble the Indiana Jones hat, or the "Smokey the Bear" style hat associated with the First World War. But even those had drawbacks, as soldiers in the west found them hot and heavy, and repeatedly requested straw hats for campaign wear. The US Army refused to grant the request on a large scale, because the felt hats were far cheaper than straw. Economics of scale, even for a tiny army (~25k in the 1880s).

Perusing through pictures, you'll often find American soldiers in casual dress wearing straw "boater" hats when off-duty, as by the 1880s it was a popular style that would only get more so as the century came to a close. They were likely personally purchased and were never considered a part of the uniform proper, but was tolerated because of their comfort, as long as they were relatively uniform throughout a specific company or post.

Working at a historic site, I've worn most of these hats, and all of them have drawbacks. The straw boater is comfortable and practical, but a slight breeze will whip it right off your head. The campaign hat gets pretty stuffy if you're out in the sun for too long, and tends to look pretty battered and shabby after surviving a rainstorm or two. The kepi is difficult to keep in place because it's a rather shallow hat that sits high on the head, meaning that it slips around when running or moving at any speed. The shako is, actually, less bad than many others in all those respects, but it's nearly impossible to keep it on your head while lying prone, which might be part of why it was phased out in favor of the lower-profile kepi as armies moved toward a skirmish style infantry doctrine. The cocked hat's only merit, in my opinion, is that it takes a hurricane to get it off your head. Otherwise, it is hot, provides no protection from the sun, and if worn in any manner other than cocked at the perfect angle, interferes with the manual exercise.

The last paragraph is entirely my opinion, keep in mind, and I expect mileage varies significantly.

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Good sources are a little difficult to pin down specifically, but in terms of general reading that might touch on headgear you should check out

Redcoat: The British Soldier in the Age of Horse and Musket by Richard Holmes

The US Infantry: An Illustrated History, 1775-1918 by Gregory Urwin

Forty Miles a Day on Beans and Hay by Don Rickey