They aren't at all ethically exempt, keeping in mind the important distinction between law and ethics.
Without digging deeply (since this isn't the "U.S. Law Forum"), I believe the legal system in the United States deals with cases like this in court. In other words, these colleges are not explicitly protected under law to simply admit and not admit whomever they please. If someone feels they have been discriminated against, they can sue. If they win their case, the college has to pay out damages but is still probably not bound to admit them, although likely in many cases they will then feel ethically and fiscally bound to do so since a precedent has been set, allowing for an endless stream of further suits.
This appears to be a bit of a timely topic. It seems from that article that in fact, the definition of "woman" in this instance is a grey area in U.S. law, so much so that various colleges have sought and received a religous exemption so that they cannot be held culpable for denying transgender women entrance.
this only makes me wonder why there aren't Native American, First Generation Immigrant or LGBT colleges that exist for the same goal, or rather why they can't legally exist in the first place.
You're begging the question. There are in fact Native American Colleges in the U.S. As for the other groups, I would imagine the issue is more fiscal than legal.
I think what I've already said applies equally well in light of your edit, but I'll flesh that out a bit.
In the US at least, an educational institution that receives public funding cannot discriminate based on various protected classes for well played out reasons.
If transgender people clearly constituted a protected class under U.S. law, then this would be cut and dried -- but of course this is not the case. Presumably over time they could become such, but it would require either executive fiat or else a consistent string of judicial precedents (and conversely, the chances of them ever becoming such could be dispelled by the same processes).
That answers the question of why a public funded college might consider itself to be legally exempt and act accordingly. It still does not make them ethically exempt, but presuming their own understanding of ethics accords with their actions, the most feasible argument [advocacy for the devil ->] is a probably a utilitarian one that you have more or less already outlined: For every transgender person made happy by allowing admittance, a disproportionate number of existing students will be made unhappy and feel betrayed in relation to what they feel the college is. So, in the interest of the happiness of the greatest number, it is best to keep transgender people out.
This is perhaps implicit in a quote from the aforementioned article:
“What it means to be a woman isn’t static,” says Mount Holyoke President Lynn Pasquerella, who announced the admissions policy change at the college’s convocation ceremony. “Early feminists argued that reducing women to their biological functions was a foundation of women’s oppression. We don’t want to fall back on that.”
Evidently their previous excuse was one based on biology -- the claim that transgender women aren't women. However, this led to an ethical conundrum that may be over the head of a good portion of undergrads (after all, this is to do with what they have come to learn, and not necessarily what they understand already). A conservative school administration (and corresponding members of government, since we are considering the question in relation to public schools) may then regard any change as too dramatic, which is the crux of the utilitarian argument. In the Holyoke case, evidently this was trumped by a deontological argument that favours admittance for the transgendered by saying the college already stands on the idea that womanhood is not determined by biology, and they would be hypocritical to pick and chose how to apply this stance.
Conversely, the religious colleges have made their own deontological argument based on their own definition of womanhood.