It's really no wonder that this story is so fondly remembered--this is really the finest example of this era of Doctor Who. The cast, both regular and incidental, are firing on all cylinders, and more importantly they've figured out exactly how to do this kind of story. The Doctor and his companions are separated from the TARDIS quickly and elegantly, kept apart by the thinness of a brick wall and by the distance of an entire civilization.
But at the same time, this is about more than simply getting back to the TARDIS and getting the heck out of Dodge; the scene between Barbara and the Doctor, one of the best in fifty-plus years of the series, showcases what the first Doctor era did better than any other. Barbara isn't willing to simply sit back and observe history, not when she knows where it ends. She must know, as a history teacher, that the Aztec practice of human sacrifice is more than mere superstition; equally, she must be aware that history has never been kind to rulers (especially women) who attempt to use their power to change the long-established and deeply-held customs of their people. But at the same time, she knows she has a chance that no-one else will ever have again. Her final line to the Doctor--"Not Barbara, Yetaxa," is delivered with such steely determination that you can't help root for her quixotic dream.
But of course, it fails. It fails not just because of the arc of history, or because of wibbly-wobbly timey-wimey reasons...it fails because one person can't change the entire course of a civilization with a well-timed "No!" It fails because the 'victim' wants to honor the gods, and Barbara's intervention (no matter how well-intentioned) is nonetheless a condescending attempt to impose her culture on another because she thinks it's 'better'. What she's trying to do, while kinder in execution, is no less a destruction of the Aztec culture than Cortez and his successors attempted.
And more than that, it fails because it threatens the power of the priestly class. Tlotoxl is a master of realpolitik more than a true believer; he's fully aware that the rain will fall without blood. But he also knows that the Aztec military machine runs on the blood of its victims, that the sacrifices are as much a message to the empire's enemies as they are a devotion to its gods. He knows instinctively that Barbara is a threat--how could she not be? She interposes herself by her very existence between him and his gods, and his gods are all that make him who he is. And further, this is an era of the series where the past isn't presented as "just like the present, but everyone is stupid except for a chosen few who act just like us". Tlotoxl is religious, but he's by no means superstitious. He knows, from a thousand subtle and tiny cues, that Barbara is no reincarnation of a dead man. Even if she was, he'd have to destroy her...but the fact that she's a fraud and a charlatan make it not just a necessity but a duty as well. Tlotoxl is one of the best villains the series has ever produced.
And all that's just in the first episode. And all that doesn't even get into Ian's beautiful confidence as he fully recognizes that Tlotoxl is setting him up to be killed by Ixta and just doesn't give a solitary fuck because he's that badass, or the Doctor's flirtatious relationship with Cameca. That is a hell of a lot to pack into thirty minutes...small wonder, to bring it back full circle, that this one is held in such high esteem.