The subject of "would not find its way onto a resume" is "a relatively minor event".
There is no comma needed, because we are still talking about the minor event in both cases: "took place many years ago" and "would not find its way onto a resume".
The reason why "would" is used:
"Would" adds the nuance of probability (or, in the case of "would not" - improbability). What the author of the passage is saying is that the minor event is very unlikely to appear on the resume, because the event is minor and its description witty and satirical - all of these qualities are in conflict with the style and purpose of the resume. Hence, "the event would not appear on the resume".
I hope my explanation makes sense, if not - ask and I'll try to clarify further.