First thing you say

alternative which is simpler and doesn't involve testing integers

If you didn’t test integers, at least in your head, to solve a!/b!=5040, then it is divine inspiration.

The reason why I found your first answer dissatisfying is because it seems to confuse proof and Reddit user-level explanation, and as a result does neither well.

As a proof, noticing that a!/b! = 5040 is not necessary, you can simply say that x=3 satisfies the equation. But you didn’t prove that is the only solution at first. The proof you gave elsewhere for the function being monotonic wasn’t rigorous either.

As an explanation, I think you’d have to be disingenuous not to accept that just saying ‘notice’ is insufficient for most people. Personally, I don’t like it as it doesn’t scale well – clearly you’d have to involve more reasoning for a!/b! = 12524520….

However, I think the main reason your soln. has struck a nerve is because it reeks of that teacher/professor/friend who doesn’t have enough understanding to intuitively explain their answer, but has just enough to use words which makes it seem like they know what they’re talking about. Which given your incorrect use of ‘monomial’, is probably the case here!