Linguists would tell you the job of a dictionary should be to describe how the word is used, not rigidly stick to some theoretical ideal. I think calculators and tools like Wolfram Alpha should do the same with maths.

You're literally arguing that what you consider the ideal should be rigidly adhered to, though.

"How mathematicians do it is correct" is a fine enough sentiment, but conveniently ignores that mathematicians do, in fact, work at WolframAlpha, and many other places that likely do it "wrong".

The examples in the video showing inline formulae that use implicit priority have two things in common that make their usage unambiguous.
First, they all are either restating, or are derived from, formulae earlier in the page that are notated unambiguously, meaning that in context there is a single correct interpretation of any ambiguity.

Second, being a published paper it has to adhere to the style guide of whatever body its published under, and as pointed out in that video, the American Mathematical Society's style guide specifies implicit priority, making it unambiguous in any of their published works. The author's preference is irrelevant.

Also, if it's universally correct and there was no ambiguity in its use among mathematicians, why specify it in the style guide at all?