Let's say a failed check represents the character making a sound a small animal wouldn't make, like snapping a mid-sized stick or clanking their armor or being a colossal idiot and saying something out loud. I think my point still stands. It's still a jump for the guard to conclude from hearing this that an invisible creature is around.
I realize you run things differently in your games, but in my games that's exactly what the guard must conclude. He knows a creature made the noise and that the noise had its origin in a particular space. Since the space seems to be empty, except perhaps for some weird footprints, the noisy creature must be invisible.
And the cloak of elvenkind still doesn't seem to offer any improvement by allowing the wearer to pose as a rock: it's not like the guard is going to conclude, "Oh, it's just that large unfamiliar rock that snapped that stick/clanked/spoke aloud -- nothing to worry about!" If anything, upon hearing this, the guard is going to start looking at human-sized rocks and other plausible hiding places very suspiciously, even if in fact the source of the sound was an invisible creature standing in the open. A hidden threat is more likely than an invisible threat.
The advantage of the cloak, in my opinion, is what happens when the guard tries to verify visually that there is a hidden creature present at the source of the sound. Let's say the guard's Perception score is 12, but the invisible creature only rolls 11 on its Stealth check. This is described as the creature losing its footing and kicking some loose stones, revealing its precise location. The guard trains his eyes in the direction of the falling stones, expecting to see the creature that made the noise, but sees nothing, which alerts him to the presence of an invisible foe.
On the other hand, the creature wearing a cloak of elven-kind rolls with advantage, resulting in a total of 14. The discarded roll was 7, so the same kicking of stones is described, but when the guard looks in the direction of the cascade of rocks at what he assumes to be the form of a hidden creature, he finds he is mistaken and that it is only a gnarled tree stump. He becomes uncertain of the direction from which the noise came and looks elsewhere for its source, since tree stumps don't usually miss their footing and cause stonefalls.
Basically, the guard is expecting to see a creature in a particular space. If the space looks empty, it is still possible the creature is there, only invisible. If the space is inhabited by a tree stump, then the creature cannot be there and must be elsewhere since two objects cannot inhabit the same space.