For the record, when during my research for 11,000 Years Lost I would refer to "thunderbirds" to archeologists they always understood me (correctly) to mean teratorns. The teratorn was a scavenger larger than the California Condor which died out during the "megafaunal extinction" period at the end of the Pleistocene. (The quotes are because not all the extinctions were of megafauna, though most were, and I'm pushing my theory that examining the microfaunal extinctions would be a fruitful line of research.) American Thunderbird myths can be plausibly sourced in folk memories of teratorns reinforced by fossil remains of flying creatures such as pterosaurs. I'm posting illegally from work, so I don't have the reference to hand, but I own a book discussing the relationship between fossils and Indian traditions at home and I'll cite it tonight for those interested. Fossil Myths of the Native Americans or something of the sort.
The Texas woman's story of the child attacked by a vulture-like bird sounds exactly similar to stories circulating during the 1970s "big bird" flap, only (if I read the post correctly) several years pre-dating it. Can you remember where in Texas she was from? Vultures - turkey and black - are the most often-seen large birds, but they are not normally regarded as nuisance birds or dangerous and the behavior described is not typical of them. Vultures ignore humans unless they're threatening or appetizing, and they don't look appetizing till they're dead - vultures cannot digest fresh meat. A threatened vulture defends himself by projectile vomiting, not by pecking or grabbing, neither of which (with his relatively weak feet and beak designed for tearing well-rotted meat) is nearly as effective as a good strong well-aimed barf of half-digested carrion. You can imagine how much fun this makes ringing them!
The point is that, if the story is true (which I'm willing to assume for the moment, though I must point out that memory is fallible, everyone will tell tall tales under the right circumstances, and if you tracked down the sister you might find that it "really" happened to a friend of her's) it is highly anomalous. A vulture couldn't physically do it and wouldn't want to do it to begin with. As the knight said, it's not a question of where he grips it - it's a simple matter of weight ratios.
You can't, however, convince some people of that; and it must be admitted that a sudden confrontation with a large bird can exagerrate the size of the bird enormously. I have a friend who claimed to have met a Great Horned Owl easily big enough to have carried him off when he was a child - it flew up as he was passing and scared him, and has been magnified in memory. Or it might have been a fairy owl - I wasn't there.
As a birdwatcher, I'm intimately familiar with the difficulty of judging bird sizes without a reference point. Quite familiar birds and animals can seem huge if they're close enough or unexpected enough. Also, once the bird is out of the eye, and you start scrambling through your references to ID it, you begin to distort the field marks. Well, I do, and I project this onto other people, but I know how easy it is, especially if the bird didn't give you a good angle or there's something funky about the lighting. Pelicans (a bird which fairly shouts: "I'm descended from dinosaurs!") change size shape constantly in flight as they teeter on the air currents at eye-straining distance. My husband has seen a black spoonbill (probably a lighting effect). I once saw an archeopteryx in New Orleans (it turned into a blue jay when I stared at it).
These sorts of things have to be taken into account when interpreting a story; but they don't, in and of themselves, prove anything about a particular experience. One thunderbird sighting could well be of an anomalous creature and another one of whooping crane in unexpected context. Unlike reports of anomalous mammals, however, I am pretty sure that there's no breeding population of enormous birds in the United States. Birds that size don't care who sees them and have to range widely to get enough food to live. Because of the relative ease with which birds can be spotted, ornithology is one of the best-covered subjects in American biology, and gets an additional boost, post-Peterson, by the popularity of birdwatching as a hobby. They would be known to science by now.
But hey, I thought the ivory-bill was extinct. If I was wrong about that I could be wrong here; or for all I know they could spend most of their time in remote areas of Canada.