The Wiki article on this is surprisingly good. It lists an early occurrence of the ballad in a published English broadside dating to 1594, so this is very old. You are absolutely correct to note the peculiar European tradition of telling or singing horrible, terrifying stories to little children. I don't know how we can account for it other than to state that this is simply part of an established, time-honored tradition.

The phrase "Babes in the Wood," entered into the American lexicon through a peculiar mutation thanks to a nineteenth-century comic. “Babes in the Woods” became famous as the title of a series of lectures given by one of the world’s first stand-up comics, Charles Farrar Browne (1834-1867), whose stage name was Artemus Ward. His lecture was a series of absurdities, not the least of which was that he only mentioned the "Babes in the Wood" at the very end of his lecture. The concluding remarks by Artemus Ward, as cited by his biographer Don Carlos Seitz, reads as follows: “But I suppose that you want to hear something about the children in the wood. They were good children, they were unfortunate, and as far as I have been able to ascertain, entirely respectable, but I found the subject such a difficult one to write about that I concluded that I would not deliver a lecture on the children this evening. I may do so some time, however, and so, in bidding you good night, I will take the liberty of saying ‘to be continued.’”

In part because of the lullaby but also thanks to Ward, the phrase “Babes in the Wood” came to refer to people who could be suckered into just about anything, taken advantage of, and used terribly. Part of the humor of Ward's lecture - and the title of his lecture - is that he had suckered his audience into the hall, presumably because they would want to hear about the "Babes in the Wood," and of course, the comic failed to deliver, making them all "Babes in the Wood."

edited to correct date of birth for Browne - thanks to /u/chimney3

Correct me if I'm wrong but isn't this a reference to leaving mentally or physically handicapped babies in the woods to die instead of being raised?

My grandmother used to sing this song to me during car trips

There was an old woman and she lived in the woods, weile weile waile. There was an old woman and she lived in the woods, down by the river Saile.

She had a baby three months old, weile weile waile. She had a baby three months old, down by the river Saile.

She had a penknife, long and sharp, weile weile waile. She had a penknife, long and sharp, down by the river Saile.

She stuck the penknife in the baby’s heart, weile weile waile. She stuck the penknife in the baby’s heart, down by the river Saile.

There were three loud knocks come a’knocking on the door, weile weile waile. There were three loud knocks come a’knocking on the door, down by the river Saile.

There were two policemen and a Special Branch man, weile weile waile. There were two policemen and a Special Branch man, down by the river Saile.

They took her away and they put her in the jail, weile weile waile. They took her away and they put her in the jail, down by the river Saile.

They put a rope around her neck, weile weile waile. They put a rope around her neck, down by the river Saile.

They pulled the rope and she got hung, weile weile waile. They pulled the rope and she got hung, down by the river Saile.

And that was the end of the woman in the woods, weile weile waile. And that was the end of the baby too, down by the river Saile.

No one will ever see this but I thought I'd mention that I took a children's lit course that began with fairy tales, and the Prof. said most of these began as morality tales: "little girl, don't let yourself be tempted by the big bad wolf (read: seduced by a man before marriage) or you'll perish;" "obey your parents or they might decided to trick you into the woods and leave you for dead!" They were warnings about the consequences of not following social conventions. Over time they were rewritten and changed depending on the time period and bent of the writer, but usually if you look at the core story there's some kind of underlying social lesson.