No, it's not.

When lifting the other character you are either taking an action, or you interacting with an object -- depending on your DM's rulings.

If the DM rules that the pile driver is an attack (which I'd argue it is), that would cost you an action, and you can only repeat it if you have movement and an action to spend on it. This assumes you've grappled them, and have an attack left.

If the DM rules you are use an object. If it wasn't an enemy, you'd only be allowed to pick it up once. Once you drop it (which is free), it is no longer a free action to pick it up again. Use An Object action:

You normally interact with an object while doing something else, such as when you draw a sword as part of an attack. When an object requires your action for its use, you take the Use an Object action. This action is also useful when you want to interact with more than one object on your turn. (PHB 193)

Since you can't lift someone and do something else, it might take a full action to lift someone above your head. Remember that a round is 6 seconds, dropping someone more than once (or twice if you have the actions for it) would take about that long. Moreover, really can't lift someone and jump while doing something else, so it could be argued it can't be an interaction and must be a "Use Object" action.

To rule otherwise would allow a free 5d6 damage without using any action, and that, while awesome the first time (and a DM may allow it once, by Rule of Cool), is game breaking.

You must release the target to attack, probably ending the grapple

D&D 5e has a broad intent that any word which is not explicitly given special rules meaning, instead has its plain English meaning, though I don't have a citation prepared for it. In plain English we see this:

Using at least one free hand, you try to seize the target ... you can release the target whenever you like

While "release" technically has multiple meanings, I find this context unambiguous in having Release mean "stop holding with your hand". From this plain English reading, I think it's clear that between the time that you "seize the target" and the time that you "release the target" your hand is occupied by that target; in other words, in order to use both your hands for a 2-handed weapon attack or other purpose you must release the target.

There does remain some ambiguity as to whether that actually ends the condition though. The sentence in question states:

The condition specifies the things that end it, and you can release the target whenever you like

But there isn't a strict connection indicating that "release the target" is an additional thing, not part of the set specified by the condition, which ends the grapple. It does seem likely that it was intended to be so based on the juxtaposition of the concepts, and supported by the subsequent section which unambiguously states that the means of ending a grapple are not entirely restricted to those specified by the condition.

Regarding "Common sense"

In a comment you mentioned that you don't bring real-world common sense into D&D, which is fine. Sometimes the rules explicitly contradict reality; after all you can't complain about someone casting a fireball on account of "magic isn't real". However, D&D also isn't a computer Rules Enforcement System where ambiguous edge cases have a single answer that's always correct and non-negotiable. Instead, D&D is adjudicated by Some Guy who, in my experience, is much more likely to be swayed by "It doesn't make sense that my Half-Orc Barbarian is physically incapable of throwing a gnome" than by "the rules don't say that letting go stops the grapple"; your mileage may vary depending on the DM. In general, I would say that ambiguous situations not clearly covered by rules are more likely to take whichever interpretation most closely matches common sense even if other times common sense is ignored.