Some hats did have straps, leather or metal, and some didn't. At the end of the period, most, if not all, did have them. But Napoleonic soldiers simply weren't really supposed to make sudden movements and charges didn't happen too frequently. Well, at least not frequently enough in order for the War Departments to make clothing more practical.
You must understand that uniforms were all about making men stand out, make them proud to wear it, and all that bla-bla. The men who designed the uniforms weren't the ones who wore them on campaign, and they had very little notion on the comfort of the regular men. One of the more famous items that choose practically above looks was the
Pokalem which, due to it's rather silly look, was abolished within 2 years of its introduction.