On the image from Nick Craver’s tweet where the screenshot of my post is displayed I noticed a clock icon on the left hand side of the post below the favorite star icon.
What’s the purpose of this icon? Is this the result of some userscript or moderation privilege? Maybe a leak of the site’s future functionality?
A similar icon is used in the iOS-app for Recently Viewed posts:
It's just a link to the timeline of the post, which used to be shown only to ♦ moderators but since 2020 it's visible for everyone.
So one of your guesses was right: it's a moderator privilege, which Nick has on all sites.
Just to clarify Glorfindel's answer: post timelines are not ♦-only although the clock-button is ♦-only. They're available for the public albeit there's no visible button under each post (for non-moderators). This is primarily because it's still an experimental feature and the feature-request for adding a visible link to the timeline under every post was status-deferred.
For what it's worth, you can construct the timeline for any post manually. For example, your question's URL is https://meta.stackexchange.com/questions/32738, and the corresponding timeline URL is https://meta.stackexchange.com/posts/32738/timeline (the page is different from users with different privilege levels; for instance, mods can view deleted comments and answers in the timeline, whereas non-mod 101 rep users can't).
I personally find the timeline feature quite helpful to track the daily vote totals, and the chronological sequence of answers, edits, close votes and comments even on the sites which I do not moderate. So if you want a timeline button under each post, you could install Stack Overflow Extras (SOX) and enable the "Add timeline and revision links to the bottom of each post for quick access to them" feature. Alternatively, there's this script.
