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.

Your comments have a notification on this site. But in my experience, sometimes it does not work as we expect. In this case, I have not come here for 2 days, so I don't check your answer out. Sorry!! And right now I will check it out!!^^

I'm not sure if you get notifications when I comment on someone's answer, but I am commenting here so you get a notification (just in case) and can see that I commented on Antara's answer.