Nope, few are, actually. Some have the talent, but not the time. Others just aren't interested.
Are Players who never DM moochers?
Sort of, but there are some folks who just probably shouldn't DM, for a variety of reasons.
Are DMs who never let others DM powermongers?
Maybe, but maybe that's just the role they prefer. I know that I sometimes lose my temper as a player (and I hate it when that happens), but that almost never happens as a DM.
Does DMing make one a better Player?
I think it can, but it's not a 100% relationship.
Does playing a PC make one a better DM?
I think maybe it's important for all DMs to act as players on occasion, to make sure they remember what it's like on that side of the table.