For the normal casting of Magic Circle, it follows the target and people only gain the benefits of the spell if they are within 10 feet of the caster. The protection is an "emanation" which means a continuously radiating effect. Other emanations are detect magic, zone of truth, silence, etc. - none of these affect things that have been affected and then left the area.
Yes, the emanation follows the target as it moves. Again, that's the way all these spells work (silence targeted on a creature being the canonical example). Nothing about the spell (or other spells with the same stats) says it doesn't move with the targeted creature.
For the "binding" version, no, the drawn circle does not follow the target - it's a physical circle drawn on the floor, not part of the spell effect. The whole point of the second version is to keep something from moving.
Really the second use of Magic Circle is an example of poor spell design; it's actually a completely separate spell that overrides a lot of the normal spell stat block (like you don't have to touch the summoned creature, the spell text makes it clear that it gets applied 'for free' with the summon/bind)...
The best way to judge what classes should have access to a spell is to compare it to the kind of spells those classes already have.
Wizard and Sorcerer compulsions tend to be forceful in method and drastic in effect: The examples you gave describe subverting the victim's normal mental process by forcibly inserting a foreign thought, and blatantly stealing control of the victim's mind.
Clerical spells, on the other hand, tend to use more straightforward methods of achieving their ends, and often express the role of clerics as spiritual leaders: Thus, the spells that allow them to speak with authority and be obeyed.
As for why the power of Power X isn't compelling... Well, if you look at the ecology of gods as described in, for example, the Planescape setting, belief is the food of the gods - without it, they lose strength and eventually starve to death, or at least enter a state that is for most practical purposes equivalent to death. A follower who only vaguely follows their patron deity's teachings is still more faithful, and therefore more valuable, than an unbeliever magically compelled to serve. Of course, individual gods might have a thing for magical compulsion, but that's not enough for something to get included in the standard priesthood package.
Oh, and like mxyzplk suggests, there's more game design-y reasons for it, too: if Clerics were good at all forms of magic, they'd be far superior to wizards, even enchanters - and playing an enchanter would be much less interesting.
Best Answer
Yes, there is a potion of Magic Circle against Evil in the DMG.
It would work fine with the first kind of casting (the emanation of protection vs evil) and would move with the target as is normal for the spell.
If you made a "second kind of casting" MCvE potion, it would still work like the spell, go back and read it carefully - it would need to be drunk by a spellcaster one round away from casting a summoning/binding spell, not by the prospective bound creature. The second kind of casting overrides several of the spell's normal parameters in its description. It might be helpful in terms of not depleting a mage's third level spell slot right before summoning something touchy, but not as a "bad guy trap."