The point you make is a good one. Pre-pubescent feelings are in turmoil at the best of times, and I don't pretend to know what the answer is to solving the puberty/trans dilemma.
As far as the general population goes it is as much fear as discomfort. Fear of the unknown and something not fully understood. There is also a real fear from natural-born women that a minority of men are trying to inveigle themselves into female society.
There is also fear and discomfort from "straight" men who find themselves attracted to transgender women and fear they may be gay. This sadly can lead to them striking out, sometimes physically in defence of their feelings.
The challenge with transgender issues is much the same as faced by racism and gay issues in the past. Martin Luther King died in 1968, over 50 years ago, and the world is still beset by racism. I like to think I'm an optimist, but I fear that transgenders have the same uphill battle to succeed.