I think the key lies with Mirri Maz Duur, the only actual sorceress at that pyre. Dany is not a sorceress. She knows practically nothing about magic. She has not studied magic; none of her ancestors, from Aegon the Conqueror on down, are described as sorcerers (the exception, Bloodraven, seems to have gained his greenseer powers from the Blackwoods, not the Targs); and every other time she comes into contact with magic before and after that pyre, that magic has derived from somebody else (the Undying, Quaithe, etc.). Everything Dany "knows" about magic comes from a brief conversation with Mirri Maz Duur, her enemy, who claims to have studied magic for years (not . . . minutes, like Dany). We never see Dany "performing" any actual magic before or after that pyre, which is weird, given the way all the other magic-users' powers grow as the story progresses (the Starks pop to mind). So do we take our unreliable narrator's word for it and assume Dany survived that fire unscathed because she's "blood of the dragon"? I think an examination of the sequence of events leading up to Dany walking into the flames points in another direction.
Look at what actually happens on that pyre. Dany doesn't just light it up and immediately walk into the flames:
Interesting. The fire was very explicitly "too hot" for Daenerys Targaryen "to bear" when she lit it, so she "stepped backward", away from the fire. Immediately after Dany backs away from the fire, unable to bear the heat,
Mirri Maz Duur is a witch, with proven magical abilities. Compare what she's doing on the pyre with what she did in Drogo's tent:
nothing human . . . The sound of Mirri Maz Duur's voice was like a funeral dirge.
When we saw Mirri use magic in Drogo's tent, she was very explicitly singing. On the pyre, she was singing as if her life depended on it, singing even as the fire swept over her. And it's only after this, after Mirri sings, after the fire reaches Mirri and starts burning her, that Dany shows any resistance to the flames whatsoever:
How interesting. The fire that was "too hot to bear" before Mirri starts her singing, before Mirri starts to burn . . . suddenly stops being too hot for Dany and her "blood of the dragon" to bear only after Mirri Maz Duur, the witch who sings her spells, who we've seen using a type of magic (shadow magic) that at least one other sorceress (Melisandre) has associated with fire, sings a song and burns alive.
And it's only after Mirri falls silent (and presumably dies) that Dany starts walking into the flames without being harmed.
This is not a sequence of events that, in my eyes, points to anything about Dany being the magical key to her fire resistance on the pyre. We saw this entire sequence of events solely through Dany's eyes. She sees survivng the pyre unscathed as her crowning moment of glory, her great vindication, a testament to her blood. She notes MIrri's song and fiery death, but as if from a distance---she's far more concerned with sililoquizing at great length about the wonderful, beautiful flames and about how she totally knows more than Mirri, a woman who's actually studied and worked at magic. If we'd seen this event through Mirri's eyes, would our understanding of what was actually going on here alter? I suspect it would.
When stories are told about witches being burned, the question people always ask is "Well, if she was really a witch, why didn't she just cast a spell that made herself immune to the fire?" Logically, if you have magic powers and some crazy person is trying to burn you alive, you'd try to use those magic powers to protect yourself from the flames. I think this is exactly what Mirri was trying to do on that pyre. When the fire comes, she's singing as if her life depends on it----and perhaps the whole point is that it did.
Yet why did Mirri burn while Dany didn't? Mirri is desperately singing even as the flames reach her body, and Dany even notes that the agony from the flames seems to be affecting Mirri's song. Look at the sequence of events Dany describes: notice how it's only after the flames reach Mirri, and her song is altered by her agony, that Dany starts experiencing the first glimmers of heat/fire immunity. Did Mirri get to finish her spell before the flames reached her and killed her? Did the agony from the flames cause her to accidentally miscast her spell?
If we look at Dany's fire immunity on the pyre, not as Dany's crowning moment of glory, but as the result of a botched fire-immunity spell of Mirri's, then I think several things pop into place. Why Mirri, who'd made it clear she thought Dany was a naive child, appeared to be casting a spell that ended up benefitting Dany (like Dany's later victories in Slaver's Bay, Dany benefits here not because of her own competence, but because of her enemy's incompetence). Why Dany, who's never demonstrated any magical abilities before or since, whose family members frequently died in fires, whose family funeral ritual involved cremation, and whose plot trajectory frequently involves her benefiting from other people's mistakes, is portrayed in conjunction with the outwardly "successful" harnessing of wild magic forces that Dany frankly knows nothing about. It's been a major theme thus far that people who try to save themselves by destroying others inadvertently destroy themselves, and that those who use fire/blood magic are not inherently immune to its effects (look at what nearly happened to Melisandre when Rattleshirt was burned, look at the fate of Valyria). I see no reason why Mirri Maz Duur would be exempt from the same karmic reckoning.
And this idea---that Dany's single instance of fire immunity really derived from a botched spell cast by Mirri---also brings up a point I haven't really seen discussed: namely, why did Dany's hair burn off at all? Hair burns easier even than flesh, and this obviously wasn't a case where there simply wasn't enough heat present to ignite hair. At first I thought this could be an issue where the magic in the pyre only protected "living" tissue, but if that were true, Dany's nails should have burned off along with her hair, since nails are also dead tissue---yet there's no indication that happened. But if we look at Dany's fire immunity on the pyre, not as evidence of some magic inherent in inbred Targ blood, but as evidence of Mirri Maz Duur botching a spell by trying to make herself fireproof and accidentally making Dany only mostly fireproof, the fact that Dany's hair burned off can be explained as evidence that the spell itself wasn't working properly.
(As a side note, I've seen the idea floated that Mirri was actually trying to hatch those eggs, that she wanted Dany to do what she did, but I think that's unlikely. Mirri didn't quietly acquiese to Dany's plan, she tried to get Dany to untie her. Mirri's whole spiel to Dany was basically "You're a child, you have no idea what you're doing, you don't understand the consequences of your actions and you're playing with forces you don't understand." Given that Dany, by attributing the pyre to some special qualities of her own rather than to the actual sorceress she sees apparently casting a spell, and especially through pretty much all of her later actions in Slaver's Bay, seems have proven Mirri right on all accounts, I sincerely doubt Mirri wanted Dany of all people to have access to three fire-breathing death machines, let alone that Mirri would have willingly sacrificed herself so Dany could hatch those eggs.)
And thematically, I think Dany being "the Unburnt" on the pyre looks less like Dany being "rewarded" for being an inbred Targ and more like Mirri being "punished" for messing with forces even she didn't fully understand.