After watching EP 19 and listening to the post-show discussion about the nature of invisibility in 5e, I decided to check out my copy of the PHB to clarify the way it worked.
Page 254: The spell "Invisibility"
"A creature you touch becomes invisible until the spell ends"
This refers to the condition "invisible"
Page 291: The condition "Invisible"
"An invisible creature is impossible to see without the aid of magic or a special sense. For the purpose of hiding, the creature is heavily obscured. The creature's location can be detected by any noise it makes or any tracks it leaves. Attack rolls against the creature have disadvantage, and the creature's attack rolls have advantage."
This could be interpreted in the way Adam and the crew have; that all invisibility does gives the attacker disadvantage. However...
Page 194: "Unseen Attackers and Targets"
"Combatants often try to escape their foes' notice by hiding, casting the invisibility spell, or lurking in darkness. When you attack a target tat you can't see, you have disadvantage on the attack roll. This is true whether you're guessing the target's location or you're targeting a creature you can hear but not see. If the target isn't in the location you targeted , you automatically miss, but the DM typically just says the attack missed, not whether you guessed the target's location correctly."
In other words, the effect of invisibility is twofold. Firstly, the attacking creature needs to pick the correct location to target. If they fail to do this and swing at an empty square on the grid, they automatically miss. Intelligent enemies might swing at the square they last saw the opponent in of course, though this is up to the DMs judgement.
The disadvantage is only a secondary effect in this case. If the enemy does manage to target the correct square, they still have disadvantage just due to not knowing the precise position of the foe within that 5ft. square.
To conclude, the PHB seems to imply a stronger invisibility than Adam and the crew have been playing with. If I've misinterpreted something that anyone notices let me know, but I thought it would be helpful to point out these passages so that the players can use correct rules in the future.
Hope people found this helpful and interesting!