My old Webster's defines the prefix trans- in various ways: on the other side of, to the other side of, over, across, through, so as to change thoroughly, above and beyond.
Terms like sex, gender, male, and female can be very problematic. So please forgive me if I'm not using the terms in the same way you do. Or, indeed, the same way the experts do. Many use the term "male" to refer to sex organs only, but that makes it difficult to talk about brain sex or gender in meaningful ways.
When I was pre-transition, part of my body was female and the other part, my brain, had at least some male elements--in my gender identification, if nothing else. In this respect, I could truly call myself transgender because my own body was an amalgam that went above and beyond the sex/gender binary.
I like to say that I have always been a boy, always been male, but that's in reference to the gender identification in my brain. My body did not follow suit.
Now that I am sort of post-transition--I'm presenting as male, accepted as male, and legally male--I still have to think about hysto and possibly bottom surgery. This makes me different from a guy who has had both, and it makes me not quite analogous to an MTF who has had bottom surgery.
What we transsexuals all seem to have in common, pre-op and post-op and non-op and everyone in the middle, is that we were raised in a gender that is inconsistent with our identification, and that our brains and bodies still bear the indelible imprint of our originally having the wrong hormones and sexual equipment. This includes the influence of hormones on the brain.
Therefore, I feel that if we use the term "transgender" in the broadest sense, we can successfully apply it to any transsexual, even one who is completely post-op.
The problem here is that some people don't identify themselves that way, perhaps because we don't usually use "transgender" in such a broad sense. What if a person I would normally label as transgender says that she is manifestly NOT transgender? Once I find out that she feels this way, I would be bound to honor her wishes and try to remember not to identify her that way. However, I might privately still think of her that way even if I never say it out loud. Why? Because she has a trans history. So how, if the situation arises, should I accurately identify that trans history without using the word "transgender"? I certainly don't want to offend her or label her in a way that she finds offensive. And some post-ops object to the term "transsexual." I can see why they do, but how do I acknowledge their trans history without being offensive to them?
The umbrella term can be very useful. I'm well aware that not everyone likes it or uses it. But would they feel more comfortable with it if it were broader? Or will it always be problematic? What if we think of the prefix "trans-" as meaning "through" or "above and beyond"? Does that make the term "transgender" more palatable, or does it just muddy the water?
Do we need an umbrella term at all? (I personally like it, but that's just me...)
