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 use of the word "can" implies choice.

Any other time the word "can" appears in a ruling, it means that the thing is possible if you want.

Take the Combat rules for example:

On your turn, you can move a distance up to your speed and take one action.

You can forgo moving, taking an action, or doing anything at all on your turn.

You can take only one bonus action on your turn

Your turn can include a variety of flourishes that require neither your action nor your move.

You can communicate however you are able, through brief utterances and gestures, as you take your turn.

You can also interact with one object or feature of the environment for free, during either your move or your action. For example, you could open a door during your move as you stride toward a foe, or you could draw your weapon as part of the same action you use to attack.

If you want to interact with a second object, you need to use your action. Some magic items and other special objects always require an action to use, as stated in their descriptions.

All these things are optional. You do not have to move up to your speed, but you can if you want. You do not have to forgo moving or taking an action, but you can if you want.

Note that it does not say "If you want to interact with a second object, you can use your action, it says you need to use your action. This is because there is no choice, you must use your action.

Leomund was a smart guy

Given Leomund's history, I don't think he would leave such a glaring oversight in his spell. It would make the spell largely redundant, as you could simply remove the earth under the hut and the occupants would walk through.

As shown in this article by Nerdarchy, Leomund was a smart guy. I don't really believe he would design a spell like this with such a large flaw in its design.

I would say the biggest reason Leomund is one of the famous wizards of distinction is because of his inventiveness. From the number of new spells he created to their utility in boosting the survivability of an adventure, Leomund had an answer.

I refuse to believe that such a clever and inventive Archmage would create a spell with such a simple flaw. I think he can design a working floor.

Conclusion

I would personally rule that the player can choose whether or not to pass through it, since the wording of the spell is in line with that and that would be best for my group. I also think it makes more logical sense if they are given the choice to pass through or not, since if you just passed through the bottom of the Hut, the floor is purposeless. There's no point to it if you have to phase through it and sit on the ground anyways.

TL;DR: Yes, they can pass through the floor if they want or they can choose to not pass through the floor and collide with it if they want.

However, I would also say that each DM should do whatever they think is the most fun for their group.