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

Solution 1:

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 (as at 05 Aug 2020).

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.