I don't think any one person is qualified to speak on behalf of an entire gender, or even of an entire gender within a single society.
However, if you must have identity spokespeople, I don't see why a transgendered person would be any better or worse than a woman. After all, they may have more experience of the difference between the two genders than a woman does. And it's not like a woman, when speaking about the experience of woman, has a useful point of comparison.
As a (cis) man I can share my experiences about being a man, but they're also experiences about being me. I don't know what it's like to be a woman, so, by extension, I don't know what it's like to be a man either. I have nothing on which to base the difference. Either I can understand enough about woman to get an idea of what it would like to be one, or I can't. If I can't, then I don't know what it's like to not be a man either, and thus I can't tell you much about what difference being a man makes. I can work out an idea, sure, but so can anyone.
So if you're going to have a spokesperson, they might as well be transgender. They've probably thought longer and harder about gender issues than anyone, they have some experience of both sides of the fence on some issues, and in practice they'll be using the same third party stats and reported experience that anyone else would in the same position.