As this is UA playtest material, it's possible there will be weird interactions with the more well-tested rules.
Having said that, I think the text of Unseen Servant is what causes the problem. The reference to 60' in the body of the text is obviously(?) a reference to the 60' range of the spell. If it had been worded "move it out of range of you", then there would be no ambiguity in the case of its range being increased (by this UA or any other feature).
As you need DM's approval to use UA rules anyway, I suggest you ask them what happens in this case. If it was me, I would increase all "60 feet" references in the spell to "1 mile" when this feature is used.
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.
Best Answer
It's spelled out fairly clearly in the spell description. The description provides a straightforward test to answer all such questions. An Unseen Servant can perform simple tasks that a human servant could do. A human servant cannot levitate, therefore neither can the Unseen Servant.
Moreover, the intent of the spell is quite clear from the tasks described. The tasks are fairly complicated - mending, serving food, and starting a fire all require some modicum of skill. But they all are consistent with what might be demanded of an ordinary manservant. None of the possible tasks suggest any physical abilities (such as levitation) beyond those of an ordinary human.
In sum, the spell creates exactly what it says, an invisible servant.