Why don't the Weasleys have a House Elf?

Let's talk about House Elves for a moment. I'm currently re-reading The Goblet of Fire which, of course, has a distinct HE subplot, and it's gotten me thinking.

The main question that keeps popping into my mind is this one: Why don't the Weasleys have a House Elf?

There is a second question, which is also floating around in my mind, but perhaps I'll deal with that another time. The second question? If House Elves are generally solitary creatures (one per household -- both the Malfoys and the Crouches appear to have only one House Elf each, despite their great wealth), how do House Elves have children?

I mean, from what we've learned, House Elves can do nothing without first getting permission from their Masters. Further, House Elves appear to come in two biological sexes (male and female, Dobby and Winky) which would lead one to imagine (perhaps incorrectly) that it takes one House Elf of each sex to create a child.

So, if House Elves are generally allocated one per great house, and you need to have two House Elves (one of each sex) to make a baby House Elf, does a House Elf approach her Master and say, "Master, may I go visit the nearby great house to visit my partner House Elf for the purposes of making a baby House Elf so you have someone to replace me when I die?"

And, if that's the way it works, to which great house is the resulting baby Elf attached?

But more on that another time (perhaps).

My first question is, why don't the Weasleys, with all those children, have a House Elf?

Part of the answer appears to be provided in my consideration of the second question above: on several occasions in the early books, J.K. Rowling suggests that House Elves are attached to great houses, like those owned by the Malfoys and the Crouches, and the families that inhabit them.

We could argue that a House Elf can only live in a manor or other great building (such as Hogwarts). That would certainly help to explain why the Weasleys don't have an Elf -- the Burrow is simply not grand enough to host such a creature.

When you read that, did your mind go where mine went? But but but but... the Black family does not own a glorious manor house and they have enjoyed the services of Kreacher and his family for generations. The Blacks live in a row house in London, not tiny nor inexpensive by any means, but not a manor house.

So perhaps you don't have to own a magnificent home (or building) to have a House Elf.

Could it be that House Elves only serve great wizarding families with long, glorious histories? No matter where they live at the moment?

That's possible, but it doesn't explain why the Weasleys (or the Gaunts for that matter) don't have an Elf. The Weasleys are well-established as a pure-blood wizarding family and the Gaunts, for certain, are pure-blood with a glorious past.

My best explanation is that House Elves formed relationships with wizarding families long ago in the distant past. They naturally gravitated to families with large homes and lots of money. Once the bond was formed, it continued throughout history, no matter what happened to the wizarding family and its fortunes.

It could be broken only by three events:

1. The wizards freed the Elf for whatever reason;
2. The last Elf in its family died, without children, leaving the wizarding family without an Elf; or
3. The wizarding family died out completely, leaving the Elf in search of a new place.

In the case of 1 or 3, the Elf would either try to find a new family or end up at a place (like Hogwarts) where it could find work and a new role in life.

In the case of 2, the family either had to live without an Elf or find a new Elf that is looking for a new place.

Think about Dobby and Winky, who find themselves, by the middle of TGOF, without a place. Dobby has been accidentally freed by the Malfoys while Winky was let go by Barty Crouch Sr. Dobby goes out looking for work at other houses but ends up at Hogwarts while Winky falls into alcoholism as a result of her shame of being a House Elf without a proper place.

If those two stories are typical of what happens when an Elf becomes free, it is surprising to me that the Weasleys were never able to pick up a House Elf at any point in their history.

It can't be their poverty that stopped them: remember, House Elves normally don't want to be paid. So what was the problem? What happened?