I DM'd 3.0, 3.5, 4e, and 5e. I can say, without hesitation, that Legendary Actions are a boon to the game.

Your player seems to resent that Action Economy does not work in his favor when hanging up on a single monster. I can say, without a doubt, based on your story, that he is being petulant.

I've run dragon fights in 3.x, and Action Economy makes those things a lot less exciting than they should be. Example: I once ran a party of 14th level characters against a CR 20 ish Red Dragon. 6 PC turns to only one for the dragon meant it was not much of a challenge. This was not only disappointing to me, as a DM (since I wanted my players to experience something epic and awesome), but anticlimactic to my players, who felt kind of jaded.

4e gave us Solo Monsters, and barring making a creature a Solo thru templates, all Solo monsters had lots of Reactions, Interrupts, and other off-turn actions. Dragons, for example, upon reaching their Bloodied threshold immediately recharge and use their breath weapon. This made encounters with a single creature way more fun and exciting for everyone. Beholders were a lot of fun, too.

5e gave us Legendary Actions. This was the inheritance of 4e's Solo monsters. It makes creatures able to challenge an entire party when encountered alone. Lair Actions, likewise. They emphasize the extra danger involved in attacking a creature in its home turf. Better to lure an Ancient Dragon into neutral ground than fight in its home.

Your player seems, to my perception, to be more interested in "winning" than "playing". I, as a player, have been in combat with Legendary creatures, and my take-away is that it was a lot of fun.