Salieri's supposed poisoning of Mozart was thus portrayed as the carrying out of a secret motive. Yet, as you point out, if they truly had been enemies, they would have written something publicly to that effect, or in their letters and diaries, or gossipy contemporaries would have recorded it in some manner.
History though provides no such evidence, an omission that most experts consider as persuasive against any claim that Salieri had a grudge against Mozart. Indeed, Salieri was a respected and successful composer, conductor, and teacher who made a large contribution to the development of the opera. While Salieri and Mozart were sometimes rivals in seeking jobs and commissions, they were also seen by contemporaries as on friendly terms.
Granted, Mozart may have been trying as a child prodigy, and his jarring liking for practical jokes and for scatological jokes and lyrics might have made him trying company at times for the older and conservative Salieri. Yet such irritants seem a thin basis on which to project a secret hatred of Mozart by Salieri and a motive to murder him. In sum, the key premise of Amadeus is bunkum.
The departures from history in Amadeus were seen by some critics at the time as an unwarranted weakening of the premise of traditional historical fiction that the story tamper as little as possible with history and be spun around the margins of but not contradict known historical facts.
In our era of mashups and take offs such objections may seem quaint but they still help mark the limits to what audiences are willing to tolerate. Amadeus was a well-told fiction and commercial success, but it was not followed by a wave of similar movies that contradict historical fact. The artistes of Hollywood usually seem to reserve such deceit for when they are trying to score political points -- and the result is often a commercial bomb.
Salieri is like Dr Doom vs Reed Richards (Mozart).