I think that's the kicker for me -- if moonbeam wasn't intended to allow for damage to occur when it is moved on the caster's turn, then there's no reason for the spell to require the caster's action to be able to move it. Contrast with flaming sphere, which allows the caster to move the sphere as a bonus action.
Apples and Oranges.
Flaming Sphere:
1) Damage on end of turn. This means that a foe only has to move away to avoid damage.
2) 2D6 damage.
3) Bonus action to attack one foe.
4) Multi-foe damage, but only on end of turn.
5) Can move 30 feet.
Moonbeam:
1) Damage on start of turn. This means that a foe will take damage if targeted.
2) 2D10 damage.
3) Action to attack one foe.
4) Single foe damage (typically, but it can be used against a horse and rider, or an enemy and it's familiar, etc.).
5) Can move 60 feet.
Wizards are supposed to do more damage than Druids with spells. Druids have better AC, hit points, healing and shape changing.
Sure, Flaming Sphere sounds better than Moonbeam and it does typically do more damage, but it's a bonus 7 average points of damage at best per round over what the wizard is already doing. The difference between 30 feet and 60 feet is actually pretty huge. The difference between doing 2D10 and 2D6 is not insignificant.
You cannot use the utility of the one spell to argue how the other spell is used.
If moonbeam doesn't damage enemies when the caster moves it on top of them, then you're saying that on the caster's turn after casting the spell, he's basically just messing with his opponents' next turns. A legendary monster could easily defeat this spell, at any level, by simply readying an action to move when the caster parks the beam over them and relying on its legendary actions to defeat the party.
Seriously? If a ready action is the solution to stopping the spell, then the spell is doing GREAT. The creature readies, the Druid does something else. The foe using up an action to bypass one spell is huge in action economy.