I don't know, but agree that it doesn't make a lot of sense. You are excatly right that we hunt white tail deer during teh rut and moose too for that matter. So why not mule deer as well? I don't know.
But I don't know a lot about mule deer in general. When is the typical mule deer rut period? Does it vary a great deal from area to area or state to state? Is it close to the usual hunting season so that there is some degree of over lap if the weather conditions are right or is it entirely in a differnt season than the white tail deer's fall rut which matches nicely with some of the earlier hunting seasons?
If you look at what type of country each is hunted in that should answer your question. Mule deer are in mostly wide open mountains type of terrain where white tails are in brushy thick cover. That along that when the rut starts there is usually quite a bit of snow that pushes the mule deer down out of the hills and into areas that are quite easy for a hunter to bag a deer if he has a tag. Some states have hunts for mule deer during the rut in November and December but the tags are limited so that you can't wipe out the population. A good friend of mine had a tag in Utah one year during the regular hunt in October and for some reason the rut was on. You could just drive around in the flats and pick the deer that you wanted to shoot. It was quite enjoyable to see them fighting and picking the one that you wanted. You have to figure that when a animal is after the other sex his thought processes are on one thing and one thing only.
It's of course easier to kill bucks during the rut, which doesn't always play into the goals of the managers. They have to factor in buck escapability models, which vary depending on when the seasons are. If your goal is to maintain a certain buck:doe ratio, and included in that may be a certain "mature" buck ratio, depending on what the standards are, a manager can only issue so many tags during that most vulnerable season. You see this especially with Colorado.
Most units issue the most rifle buck tags during 2nd season, when bucks are usually hardest to kill, then issue fewer during 3rd season, and between one half and one tenth of the 2nd season tag numbers during 4th season. Similar idea with the Eastern Plains seasons. Or take a look at Utah, they also balance hunter numbers ($$$) during the general seasons (October), then offer just a few tags during mid November or even December in some units for the trophy hunters. Wyoming is somewhat similar to Utah in this respect
Moose calling can get confusing to some, but cow calling is your main tool when moose hunting. If you go out and buy moose calls I would say you're wasting your money unless it is the birch bark cone. I find most store bought moose calls sounding worse than puffing a grunt your hands to make the call. You can roll up a piece of birch bark in a cone shape to use as a call and spend your money on a video that teaches you the basics of a cow call. A milk jug or a coffee can with a hole and a rope...