Michael Butterfield - is your colleague also a Target release user? Perhaps the anchor links will not yet work for Standard Users as the code for the heading IDs (which is what the links rely on) is not yet available in the standard release tenants.
I've discovered that you can't add anchor links to the Quick links web part - a bit disappointed by that as some of my authors can relate to that web part more readily than in-text links. Also, if they worked with Quick links, the additional styling/layout options may be possible. I agree with Michael that a dedicated Web part that authors can add to the page that detects the anchors and automatically creates the table of contents would be a good thing, and pretty much what I expected. However, I can live with the solution as delivered - hopefully it's just the minimum viable product and will be enhanced over time.
I've noticed something else in my tenant that was not mentioned in this (or the earlier) post. I now have a control that allows for the duplication of sections and web parts. It's working fine, but would be good to have some information about how Microsoft see this feature helping users.