Let's address each of those points one at a time:

Heat and Sun

It should be noted that a lot of the uniforms you see were used by European armies in colder climates. In addition, campaigns often involve camping outside at night, which could become surprisingly cold even in the summer. Layers can always be stripped off during the heat, whereas generating new coats is not really possible. Remember that most of soldiering is trudging through terrible weather and being miserable while waiting for orders. The actual firing your musket and getting shot part is a relatively small part of war.

Additionally, having personally worn a WWI era wool cavalry uniform with a ridiculously small brimmed hat in over 100 degree Los Angeles summer for a full day, it's actually not as bad as you'd think. The layers of insulation help keep the heat out and the tiny visor actually does keep the sun out of your eyes.

Thick Wool

In addition to keeping a soldier warm at night or in the rain, wool offers some protection against cuts, shrapnel and saber slashes. I remember one account of a British infantry soldier who stood as rearguard against pursuing cavalry whose uniform was sliced to ribbons by swipes from sabers, but managed to stay alive (my efforts to find where I read that account has failed me). Wool would have also been a lot cheaper for much of that period compared with cotton, since wool was available locally in Europe, whereas cotton would have to be imported, mostly from India and the American South.

Distinctive Styling and Bright Colours

There is a common misconception to the effectiveness of early firearms and the reasoning behind their use by large formations of colourfully dressed men. The average musket pre-American Civil War were smoothbore. Most of them had no sights and couldn't be reliably expected to hit a target beyond 100 yards, even with trained marksmen. Rifled muskets existed, but they were slower to load and expensive. Common doctrine at the time was to fire a few times then charge forward with the bayonet (or sabres, for earlier practitioners of this type of warfare such as the Caroleans), relying on the the enemy's reluctance to be stabbed to win the day. A volley of blackpowder muskets created a plume of smoke and was disorientating in its loudness, and you could reasonably expect to take less than a volley or two if you tried to cover the 50-100 yards between without taking too many volleys yourself, most of which will be inaccurate since the space would be filled with smoke from both armies.

In those circumstances, the bright colours, large tall helmets and distinctive dress were useful. There were no radios, so communication involved waving signals, portable instruments and a lot of yelling at men who were temporarily deaf due to musket fire. You had to keep soldiers together in groups where the volume of fire would make muskets effective, they would be easier to command, and they were less likely to desert because nobody was looking. You also wanted to be able to recognize a unit as your own or the enemy through the dust and smoke, lest you fire and charge an ally by mistake. There were numerous incidents where mistaken identities in the heat of battle cost lives and even battles. A notable incident happened during the Napoleonic Wars during the Battle of Wagram, when French units in blue fired upon their Saxon allies in white, because the enemy Austrians also wore white and could not be distinguished in the dark.