But if this medical anomaly is indeed a vestigial organ of a tail, than it means our ancestors had a tail. Which means the ancestors of Old world monkeys (the ones we have in common, of course) should also have had a tail.
Technically also, old world monkeys should also have some rare individuals with such vestigial organs.
I think there's a bit of confusion here. Old World monkeys do have tails, and our common ancestor with them did have a tail. What they don't have is prehensile tails.
If you look at the phylogeny of primates
here, the only animals in it that have prehensile tails are the New World monkeys at the top. As far as I know, all New World monkeys have this prehensile tail, while no other primate we know of, living or extinct, does. The simplest explanation would be that prehensile tails arose just once in the line leading to the New World monkeys. This means that when we're trying to explain the loss of the tail in apes, the tail we're talking about wasn't one that could grip anything.
So does the advantaged of having a tail outweigh the disadvantages of having a tail ?
We know that for our ancestors who did have a tail, it was advantageous to have a tail, or else they would not have developped one in the first place. Since we also know it was not prehensile (because of 'Ida', more info on this would be nice), the advantage was not to help them hang in trees. If this advantage was for balance (which is the only alternative I see), they would still not have lost it when becoming ground-dwelling since the tails of other ground-dwelling mammals are advantageous.
We don't know that our ancestors' tails weren't prehensile just because of
Ida - she's just one more primate. As I wrote above, it's because all the known prehensile tails are next to each other on the primate family tree, so it's just the simplest explanation that the trait only arose once.
Bipedalism can't be the explanation behind tail loss. Humans are the only fully bipedal ape, but no apes have tails - including gibbons who spend the majority of their time up in the trees (as do orangutans, I think).
What's more, apes aren't the only mammals to have lost their tails. There are rodents, sloths, hedgehogs, bears and a variety of other ground-dwelling animals without tails - clearly they aren't vital.
I don't really have any idea why apes don't have tails, so sorry if you were expecting a more definite conclusion to this post. It's just a couple of things to think about.