I don't think that's what most people mean by "swingy."
"Swingy" connotes a certain degree of randomness. Consistently dealing a lot of damage isn't "swingy," it's just deadly. Unpredictably dealing a lot of damage is more what people mean by "swingy," IME.
I'll buy into that concept.
Some of the posts in this thread point out spells that dramatically change the fight. That's what spells are for. If your spell is effectively the same as me chopping with my sword or shooting with my bow, then you don't have a better mousetrap.
So ignore the spells for a moment. Magic, by definition, short-circuits the universe to make fantastic things happen. Magic does add swinginess, but I think it is important to consider that swinginess can exist in the game's non-magical elements.
Consider the low-level non-caster party facing level appropriate humanoid monsters.
Arm the bad guys with swords like the party, and you have a fairly even fight.
Arm them with great-axes, and you have a swingy fight.
the bad guys could roll low for the whole fight (or at least, not any better than the PCs), or they could roll 11 and 12s for damage. Suddenly, those PCs are dead.
That's swingy. When the combatants have a fairly low HP count, but do a lot of damage. The outcome is mostly based on who rolled luckier
Give everyone more HP and/or lower the damage range, and you get a grindier game.
Grindy doesn't mean it is more tactical. In fact, it may mean the opposite. In a grindy fight, there's not much you can change as both sides trade blows and WHITTLE the other side down. By definition, nobody has any ability to dramatically change the fight, because everything's been watered down to grind.
In a swingy fight, some of that swinginess comes in when one side does something that can dramatically shift the situation. It could be rolling a 12 damage with the great-axe which is the luck component, it could be casting a sleep spell, taking out the enemy with 1 die roll.
I suspect an optimum design would be that some fights are swingy, some are grindy, with low level being a little more grindy than swingy, because the PCs have less resources to deal with a sudden shift (ie, 10 HP -12 damage=dead)