If the caster falls, the spell ends once the caster is no longer in the original area.

While the linked questions have some debate about whether or not the dome moves with its surroundings, the text is clear that if the caster moves out of the original "stationary" position, then the spell will end:

The spell ends if you leave its area.

Therefore, as formulated in your question, the hut would have disappeared once the original caster was no longer in that space.

I agree with the linked answers that there is no clear definition of "stationary," which leaves the crux of your question up to a DM's judgment call. As a DM, I would have ruled that the hut does not fall because there seems to be a clear "stationary" reference frame based on your description of the fight: the rest of the environment that isn't moving. It's worth noting that the range of the spell is "self," and since the hut obviously doesn't move with the caster, the location of the hut is where the caster was at the moment the spell was cast.

While other DMs could reasonably rule otherwise, it opens up a lot of judgment calls--how big does the piece of moving ground have to be? If the caster digs up the ground and moves it around, could they move their hut? In this case, where you're on solid ground, it seems easier to say that it doesn't move at all.

Leomund's Tiny Hut has a floor

However, the scenario in your question could have only happened if the players willingly fell out of the hut, because Jeremy Crawford tweeted that the intent is that it has a floor to stand on:

Leomund's tiny hut does have a floor, Mr. Crawford (read your own book). The spell's range entry says the effect is hemispherical.

Thus, even though the ground fell out from under you, you could have still stood on the floor of the hut.

As for balance concerns, many Big Bads often have some way of dispelling magic. Leomund's Tiny Hut, a 3rd level spell, would be automatically dispelled. If the gargantuan attacker were only a physical attacker, it could have simply left and waited out the spell's duration out of range of your attacks--you can't pursue it, after all, without leaving the protection of the hut.

The Leomund's tiny hut stops catapult in both directions

Leomund's tiny hut states:

Spells and other magical effects can't extend through the dome or be cast through it.

Catapult states:

The object flies in a straight line up to 90 feet in a direction you choose before falling to the ground, [...] When the object strikes something, the object and what it strikes each take 3d8 bludgeoning damage.

• The 90-foot line is the spell's AoE, and this line cannot extend through the dome.
• The object is not launched like a projectile, it flies, and this flight is an effect of the spell which cannot extend through the dome.
• The damage dealt is an effect of the spell, so this damage cannot be dealt on the other side of the dome because the spell cannot extend through it.

This is a moot point in catapult's case, but if an object was inside the dome when you cast Leomund's tiny hut, then that object can move freely through said dome, as stated:

Creatures and objects within the dome when you cast this spell can move through it freely.