RAW perfectly allowable, but the DM may want to set some conditions if they are big on verisimilitude
If I understand the OP (and I'm not sure that I do), the following argument is being made:
It is clear from the spell description that creatures using the hut to take shelter should be completely protected from melee attacks
Creatures and objects within the dome when you cast this spell can move through it freely. All other creatures and objects are barred from passing through it. (spell description)
Perhaps it is not clear whether defenders are equally protected when they deliberately stand at the edge of the dome and strike at creatures outside the dome. That is, in the process of making a melee attack, since they and their weapons pass through the dome without hindrance, could there be a point at which some part of them, an arm or a hand, passes through the dome and becomes a viable target?
what constitutes 'passing through it' as it pertains to combat?...My party was basically using the Hut as an unbreakable shield while still attacking foes directly outside within reach. Is that allowable? (original question)
RAW Perfectly Allowable
D&D is not a fine-scale combat simulator: 'movement' and 'attacks' are separate, and in general creatures don't move as they attack, even for piercing weapons which in the real world might imply a thrusting advance. Melee attacks are made on targets 'within reach'. In 'theatre of the mind', most will be attacking things 'within five feet'. On a grid map, they can attack things in adjacent squares. It doesn't matter whether the PC is a Dragonborn with a greatsword or a gnome with a dagger - they both have a 5 foot reach. RAW, a PC can stand inside the dome and attack creatures outside the dome in melee, and the creatures outside the dome cannot respond.
An extreme example could be a halfling delivering a touch-range spell as a melee attack. Her target must be outside the dome (because they can't penetrate it). Her spell can't be cast through the dome (because the dome description says it blocks spells). She wants to remain inside the dome (so that she can't be attacked in return). Is this a problem? RAW, no. Her square is "covered by the effects of the hut". Her opponent's square is not. The squares are adjacent. It's a valid target.
The DM may disagree
Rule 0 applies. The DM may decide that, to preserve verisimilitude, making a melee attack without a reach weapon on something outside the dome, or especially with a touch or natural weapon attack, requires the attacker to expose at least part of themselves, at least part of the time. What could happen then? Since we are departing RAW, anything - but here are some likely scenarios.
First, realize that exposing yourself for part of your turn doesn't necessarily expose you to a counterattack if your opponents aren't acting on your turn. "Cover" exists in space, not in time. Even if the DM requires you to 'leave the dome' to deliver a touch attack, on your turn you can leave, attack, and return - it is your turn, and your opponents can't take actions.
The exception, of course, is if they have readied actions to attack you when you expose yourself. In this case, their reaction interrupts your action. Here, the DM would need to decide how exposed you have made yourself - could they attack you with three-quarters cover? Half cover?
Finally, when you retreat back inside the dome, do your opponents get an opportunity attack? Have you 'left their reach' by moving from a place/state they can attack to one they cannot, even if you are the same distance away? Perhaps they would
Related: What are the rules for attacking a creature inside a Wall of Fire?