Okay, what is the advantage of being an elder?
Status, I guess? Good question. Personally I saw the prospect of becoming an elder as more of a threat than anything. It seemed like a punishment for being zealous. "Good job being a ministerial servant, now you have to: give more and longer talks, conduct field service more, set an even better example (or give the appearance of one, at least), handle various bureaucratic stuff and perhaps sit on judicial committees, go to elder's meetings after the public meetings are done and everyone else is going home, and deal with various and sundry problems thrown into your lap in person and over the phone at all hours of the day by the neediest publishers in the congregation."
And how many people in any given congregation want to become one?
Probably not many more than are ones. There's usually trouble finding enough brothers who are fit for promoting to elder or MS, and usually if a brother wants to qualify, he will be able to do what's needed. That's why there's often talks being given to encourage brothers to "reach out" for more "privileges", and this little speech is also given to brothers when the elders visit them on shepherding calls.
Is it considered gauche to express a desire to become an elder?
More or less, yes. But if you just say that you're "reaching out", it's a euphemistic way to say the same thing. If you say that often though, the elders will be leery of you for seeming power-hungry.
Also, if one had more ambition, how could one become a member of the Governing Board?
Well, you'd have to profess to be anointed, and you'd have to climb the ladder from elder to circuit overseer, then district overseer, then various positions in headquarters leading up to becoming one of the helpers to the Governing Body. It's probably possible to bypass the overseer rungs by taking the Bethel route and being promoted internally. Of course it would be pretty hard to actually do all this on purpose -- about the same odds as starting off as fry cook in a McDonald's and aiming to become a VP of the McDonald's corporation.