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We are a group of new players and I feel that I miss some important properties of the levitate spell as it proved extremely powerful in the last encounter in Cragmaw Castle in The Lost Mine of Phandelver.

The party caused a alarm to ring which caused the entire final group, with bodyguards, to prepare an ambush when they stormed the last room.

The Doppelgänger

managed to immediately down a player in the surprise round and after the rest of the enemies chucked away quite a bit of health as well it almost looked like it would become a full party wipe.

The wizard cast levitate on

The Doppelgänger

(and it failed the saving throw) which effectively disabled it. Thanks to the 10 minute duration and the wizard carefully saying that it "should float riiiiiight in the middle of the room" (so it couldn't grab any wall object to pull itself down) it wasn't able to do anything during the entire rest of the fight. From my understanding it was not allowed to re-try the saving throw on its next turn. The other enemies were not able to break the wizards concentration either as she was hiding behind the rest of the party in a small doorway which blocked the enemies from reaching her.

I'm happy the party managed to survive the encounter, but feel that casting levitate on any melee-based enemy could quickly become a effective, but boring, strategy. Is there anything in the rules I missed?

You didn't miss any rules; levitate can do that if used cleverly

Your party's wizard was being clever and resourceful. And your question has in it the key to your answer: the enemy did not make the saving throw.

An additional point: the party acted with good tactical sense, by preventing the wizard from taking damage and thus being subject to a concentration save. That's two smart things your wizard, and your party, did.

Something to consider: a levitated enemy can still cast spells or throw missiles, or even give orders to underlings, which a held or Tasha's laughing enemy cannot. That your party's wizard used positioning well is good.

• If the DM/monster didn't consider using thrown weapons or spells as a
  response to its predicament, tuck that into your "for future reference" folder.

Had the enemy made the saving throw your party's wizard would have spent a precious second level spell slot and Nothing Would Have Happened. This is one of those "all or nothing" risks taken during combat. Likewise, a concentration check, failed, creates a "nothing" consequence.

What you are seeing is how swingy the "save or suffer" spell family is.

A similar thing is true about the mentally, rather than physically, controlling spells: hold* person, hold monster, Tasha's Hideous Laughter, dominate, and a variety other control spells that are intended to put an enemy out of action. (Hence "battlefield control" as a thing that casters can specialize in).

That's intended - taking an opponent out of a fight to make the rest of the party's job easier - but the risk is that the enemy makes the save and the spell does exactly nothing.

Yes, a levitate can put a melee-only enemy out of action

You have already stated the relevant part of the spell:

"The target can move only by pushing or pulling against a fixed object
or surface within reach (such as a wall or a ceiling)"

So if you managed to levitate a creature into the middle of the room (with a high enough ceiling) then they will not be able to move.

Nothing stops them from using ranged attacks, spells and so on if they have them. But if their only attack is a melee weapon then they are pretty much stuck.

(Of course, PCs will not be able to attack them back without using ranged weapons or spells themselves).