In the end, when you finished your PhD and got a job as a postdoc, you did end up including your Stack Exchange participation in your CV.
More specifically, you did this as a single bullet point in the section
Science communication, under
Professional experience, which looks roughly like this:
(using a font which does not highlight the links quite so aggressively), and together with the rest of your science-outreach activity.
After a few years of using this CV, nobody has commented specifically on this part of the CV, either positively or negatively.
In general, adding PSE-specific mechanics (including your SE reputation, badges, or other quirks) feels somewhat crass, and it will not be understandable to the academic hiring panels that will be reading your CV, though the (relatively) new stat of "people reached" is intuitive enough that it makes the cut. In contrast, it feels much more natural to place the emphasis on the writing samples which took a lot of work, demonstrate in-depth knowledge of physics and good writing ability, and which the readers of the CV can appreciate.
I recently did an interview for a fellowship where the interviewer explicitly brought up my Stack Exchange participation. My application materials mentioned that I "am passionate about science communication and education, and I have authored over 2,400 science-communication pieces" but did not mention Stack Exchange by name.
The interviewer (who is a frequent contributor to a science-outreach YouTube channel) explicitly mentioned my Stack Exchange participation in a good light, and then asked whether I intend to maintain that level of public engagement during the fellowship. The results of the interview will take some time to come out, but I'm pretty confident in saying that this was considered as a positive aspect of my application materials.
That said, of course, this is a single data point. The funding organization in question places a high value on science outreach, and this probably colours their choice of interviewers as well as the criteria the interviewers are provided. In other places, your mileage will probably vary.
Since I'm writing to you from 2020: this year, make sure you're well stocked with toilet paper. Since I'm writing to you in 2014: buy some bitcoin!