Why is Tsuyu sauce for cold Soba noodles served on the side?
The simplest reason is texture. The point of cold soba noodles is to taste them at the ideal texture for soba.

In a party situation, I've served cold soba with pre-mixed sauce, and there's a variant called wanko-soba in some parts of Japan (particularly Iwate prefecture) that is often served premixed in small bowls, but unfortunately, this results in the noodles softening fairly quickly. They'll never be as good as they are when you dip a few strands at a time as you eat them.

In fact, hot soba noodles deteriorate in texture as well, but this can be partially mitigated simply by slightly undercooking the noodles, since they will continue to cook in the broth.

Since soba is generally a quick lunch in Japan, you'll notice the quality decline in a hot noodle dish, but you'll probably eat fast enough to avoid terribly soggy noodles (unless you're one of my toddlers, but even they can plow through a bowl of soba faster than they will almost any other food).

Cold noodles will get soggier, but they will not "cook", so undercooked noodles aren't practical in that case.