Why is it unacceptable to use the word "conquer"?

I believe your instructor is correct. The meaning of words change over time, and what was once acceptable may not be any longer. With that said, conquer, historically, would have been the correct word to use, but no more. That's because the word conquer when used on people has ethnocentric connotations, it is essentially choosing to relate the information on the side of the murderers (since that is what colonizers are generally accepted to be these days).

That is because the word conquer now means to overcome, in specific it's to overcome an obstacle of immense issue. That's why people will say they conquered a mountain, or a musician has conquered the world, which is hyperbole anyway. By using the word to describe people as having been "conquered" its lowering them from the status and respect they deserve as a human to that of an inanimate object, not that the earth is inanimate, but in that quote that is the implication. That's why it's not correct to use the word conquer when referring to people and societies.

Your instructor is wrong to suggest that 'conquered' is politically incorrect. The primary meaning of conquer remains to overcome or take control of somewhere, something or someone by military force. The Cambridge dictionary even uses the Spanish conquest of the New World as an example phrase[1].

To say that "Cortez conquered the Aztecs" is a short way of saying that he overcame them by military force. It does not imply any lack of respect on the part of the author, nor that the 'conquest' was a good or bad thing.

Successfully, and usually happily, overcoming an inanimate or abstract obstacle is a slightly hyperbolic derivation from this, but has not replaced the original meaning, nor has it automatically conferred positive connotations to the military sense.

For example, it is possible to say both "German forces conquered France in 1940" and "Allied forces conquered Germany in 1945" without implying that either was good or bad.

The one grey area in your particular example is that Cortez didn't really 'conquer' the Aztecs in 1519: he began his campaign against them in 1519, but did not overthrow the Aztec empire until 1521. To imply that he 'conquered' the Aztecs with less resistance than in reality might, possibly, be seen as disrespectful. Perhaps this is what your instructor meant.

[1]: https://dictionary.cambridge.org/dictionary/english/conquer as at 05 Aug 2020