Asking whether or not that should be a problem is perfectly valid. You can argue that if the characters can open with a battle-ending spell, then so be it. The response is probably that it 1) makes creatures that should be hard to kill rather easy to kill, and 2) makes high-level casters way better at slaying (and even solo slaying) than any other class.
In older editions of D&D, tougher creatures had Spell Resistance, which give a flat failure chance to any magic. It gave them an extra layer of defense against magic, but the end result is that magic just has a much lower chance of working. There's still a chance of a battle-ending spell landing on the first round, but only if the dice are on your side. That brings high-level magic users down to a more balanced chance of ending the fight quickly, but it also means that they could spend a good portion of a long fight having spell after spell whiff. Not so much fun.
5e replaced Spell Resistance with Legendary Resistance -- the choice to just decide to make a save, limited to a fixed number of uses. I don't think Legendary Resistance has a balance issue, but certainly has a fun issue -- it's just a boring/flavorless mechanic that makes save-or-die or save-or-suck spells less able to take out a boss on the first round or two. As a mechanic, from the players' point of view, it is not much fun. It is not fun to outright shut down magic spells. It doesn't matter if you use magic or not, whiffing on your turn is always disappointing, but magic spells being a limited resource, a spell getting wasted stings a lot more than weapon attacks failing to hit the target's AC. Using a precious high-level spell slot and a creature saving for no damage or effect sucks as a player. Legendary Resistance just guarantees that it's going to suck.
Legendary Resistance's advantage (over more random methods, like rolling Spell Resistance, etc.) is that it has a fixed number of uses, and after burning through the uses, you now know you can go hog-wild with big spells. Well, as wild as you can. Some creatures are going to have Magic Resistance, giving them advantage on saves, so it still can be hard to land a spell. But Spell Resistance gets around the issue of the creature being able to survive a few rounds without shutting down casters for the whole fight.
On a related note, I think it's a good idea to make any Legendary Resistances on a creature encounter just a up-front fact, so "Surprise! Legendary Resistance!" doesn't happen, because that sucks. "It's got 2 Legendary Resistances left, what middle-level spell can I cast to maybe force it to burn a resistance?" is a much more fun and tactical question to have to answer as a player.
The whole idea of making "boss" fights feel cinematic is something I think a lot of RPGs struggle with. If you want to draw on action and fantasy movies for inspiration, those final fights in cinema usually follow the same pattern: They have the heroes start out the fight heading downhill precipitously faster than the bad guy (or girl, or creature), then there's a turning point where the heroes gain the upper hand (often just in the nick of time, but sometimes a hero falls before), followed by the bad guy getting their ass handed to them. That's good, exciting fun. When that happens in a game, that's the stuff you talk about for months or years later. You don't want all your encounters to be cookie-cut from the same pattern, but most RPG mechanics make that sort of battle narrative arc really unlikely. As a GM, you can engineer stuff like special defenses the bad guy has to help force it, but that risks putting the encounter on rails.
I'd argue that Legendary Resistance actually does a good job of helping the narrative of the big battle. The last Legendary Resistance getting burned can be that turning point of the battle -- high-level caster holding back their biggest spells are now able to unleash them. D&D battles are usually front-loaded with big damage spells and special abilities all happening in the first round or two, which runs counter to the cinematic battle. For all its warts, Legendary Resistance can be a great tool to help push that back to enable the cinematic turning point.