No, because the hut is made of magical force
Unfortunately, just reading the description of the tiny hut spell itself is insufficient to answer this question. You also have to understand that the hut is made of magical force, and that objects of magical force existing on the Prime interact with the Border Ethereal to create barriers in a specific way described in the DMG, as in the question you cite (emphasis mine):
solid objects on the overlapped plane don’t hamper the movement of a creature in the Border Ethereal. The exceptions are certain magical effects (including anything made of magical force) and living beings.
If the hut is on the Prime, it is not on the Ethereal
Let's assume a hut was cast on the Prime and your caster was in it when it was cast. The rules of the spell say that your caster can move through the dome freely.
Then your caster casts Etherealness, and they are shifted to the Border Ethereal space that maps to the Prime space within the hut. It is important to understand that the Border Ethereal, but not the Deep Ethereal, maps one-to-one with the Prime it overlaps:
The Ethereal Plane is a misty, fog-bound dimension. Its "shores," called the Border Ethereal, overlap the Material Plane and the Inner Planes, so that every location on those planes has a corresponding location
on the Ethereal Plane.
Although the dome maps to a space on the Border Ethereal, the dome itself does not exist there*. Spells do what they say they do, and nothing in the spell description says that the hut exists as an object in any plane other than the one in which it is cast*.
The hut is a solid object existing on the Prime and only on the Prime. Normally, such objects don't hamper the movement of creatures on the overlapping Border Ethereal, as stated in the DMG. However, the hut, as a physical object made of magical force, is an exception to this, and in fact it does hamper movement on the Ethereal, also as stated in the DMG. The effect of the hut, but not the hut itself, acts as a barrier, and creatures on the Border Ethereal cannot pass through the effect, regardless of whether they are attempting to leave or enter, and regardless of whether they are permitted through the dome.
This is not an exception or contradiction to the spell description, which says "Creatures and objects within the dome when you cast this spell can move through it [the dome] freely." It is not a contradiction of the spell because on the Border Ethereal, your caster is not passing through the dome itself as an object. The dome is still on the Prime. Rather, the caster is being hampered by the effect of the dome in the Ethereal, and the spell makes no reference to that.
If your caster were to return to the Prime, and they were interacting with the actual dome, they would be able to pass through it freely, just as the description of the hut allows.
If it helps, consider what would happen when you cast Etherealness on a willing elemental of air, water, or fire. While on the Prime, that elemental can remain in the space of another creature also on the Prime, because of their elemental form. But once in the Border Ethereal, they are unable to remain in the space of a living creature on the prime, because the DMG passage explains that living creatures on the prime hamper movement in the Border Ethereal. It is not the creature itself with which the elemental is interacting, because that creature is on the overlapping plane, not the Ethereal. Rather, the elemental is bounded by the manifestation of that creature in the Border Ethereal. Similarly, your caster is not hampered by the dome itself, but by the effect of the dome on the Border Ethereal.
If the hut is on the Ethereal, it is not a barrier to you
Now consider a case in which your caster casts Etherealness first, at a high enough level to bring the caster of the hut with them. Once both of you are on the Border Ethereal, the hut is cast, with your caster inside it.
Your caster can now pass through the dome, as permitted by the description of the spell. The spell does not care where it is cast, but now the dome itself is on the Border Ethereal with your caster, and the spell permits you to move through the dome. Other creatures on the Border Ethereal will still be hedged out by the dome, again per its description. However, creatures on the Prime can pass through the space the dome overlaps there, because the dome itself does not exist on the Prime any more than a Prime-cast dome exists on the Ethereal. What is more, there is no rule in the DMG that objects of magical force cast on the Ethereal have effects on the overlapping Prime, only the reverse.
*Note that this is in direct contrast to the wall of force (and forcecage) spells, which specify that they exist both where they are cast and on the Ethereal Plane:
The wall also extends into the Ethereal Plane, blocking ethereal travel through the wall.
Further, they make no distinction between the Border Ethereal and Deep Ethereal, saying only that they extend into the Ethereal Plane. This likely means that they form a barrier against any Ethereal travel, whereas the hut, restricted to the Border Ethereal by the DMG, would block travel only on the 'shores' corresponding to the prime, but would permit access to something arriving from or departing to the Deep Ethereal. Even further, this may mean that it is possible to summon Leomund's Secret Chest from within Leomund's Tiny Hut, a tribute to the careful craftsmanship of esteemed mage and veteran traveler Leomund.
