And this is a very good point indeed. However I would still expect reaction along the line "so, that dwarf got a glowing hand and now a Herald of Andraste? Blasphemy!"
But that is the reaction (to almost all races, really). You're labeled a heretic and the Inquisition as heretical. And it is heretical, no matter who the Herald is, because it is basically defying the hobbled Chantry at that point.
However, if a peasant literally knows you closed the sky over his head before it could swallow Haven/The Hinterlands area and spew demons at him, then why wouldn't he be pretty willing to put aside heretics and race? How much skin does he really have in this fight?
It's the Inquisition thing. History of Andraste knew only two races - humans (mostly) and elves as allies. Other races existed in their own worlds, totally uninvolved - neither as enemies nor as allies. BW did all they could to include other races (and you already convinced me, that they succeeded with "mystical dailish, an old race coming to save the world once belonged to them") but I still would not play the Inquisition card for dwarves and kossiths.
It's not about racism, it's about not being the subject of that lore. They could include some green people from the moon - and even that would look better. There had to be another way. Anything would do - just magic (magic opened the sky - so magic will close it now and that person stood up to the evil magister), or the race-related lore (humanity got caught up in [race] conflict with some Big Evil) - anything but Andraste and Inquisition. Yes, I know, could not be done. But the way it was done is not convincing for me and looks artificial and fairytale-ish.
Since Shartan (the elf who helped Andraste) is considered a heretic, and since the Dalish are very big on their pantheon worship and couldn't care two figs for Andraste or the Maker, I don't see the Dalish elves as any better religion-wise. Maybe city elves, as many are Andrastian. You're right that the dwarves and Qunari aren't involved in the historical events of Andraste, but I don't really see the historical events as relevant. Maybe I'm missing your point.
The Inquisition card is "I have a magical glowy hand that can close these rifts in the Fade and a woman, who looks a LOT like "Andraste" seemed to watch me from the Fade." Oh, and the Right and Left hands of the Divine are totally backing me up. I'm not saying race doesn't "matter" - it clearly comes up, and it does - but I don't think it'd prevent people from recognizing you as the Herald. Being surprised and potentially unhappy (especially TPTB)? Sure. Any race but human really and even human mages to some degree. But what does some farmer or soldier or villager care about your race if you can literally perform miracles? I don't get the separation from the lore at all, really. The "lore" of Andraste - if you're Andrastian and believe in her and the Maker - is for everyone. Just because some races have not received it does not mean that the Maker doesn't have a plan for them or watch over them. Now the "Powers that Be" of the Chantry are not going to be happy with a Herald who is a) Male, b] a Mage, c) Anything but human. Because that bucks tradition and threatens their power. But those people have just had their asses handed to them and are calling for your head whether it has knife ears, horns, or not. They can't have it. They can't reach you. You have the magic hand, and you have people who recognize that and back you up.
Personally, based on the whole story, I don't believe the Inquisitior is necessarily any Herald at all (it's not impossible, but neither is it cemented) so I don't get the ,lore issue since the game doesn't really explain Andraste (any better than we know), prove her involvement, or even establish the Maker as real; all that remains a mystery. But you can't argue with Cassandra - she is what they needed, when they needed it. I think it is, likely, far more difficult to play as a devout Andrastian dwarf or Vashoth (though Varric is Andrastian by many accounts, so not impossible for a dwarf to be one). But you don't have to really be Andrastian to be the Herald. Being the Herald is about being what is needed. (Which parallels nicely with Andraste's historical story, actually, and its many accounts.)