As often happens, I found the official documentation and forum answers to be "close, but no cigar", and so had to experiment a little to get things working.
The main problem for me was a lack of concrete configuration examples. That's not entirely GitHub's fault: having migrated from Google Domains to Namecheap in the middle of this project, I was once again reminded of how many different ways there are to do things in the name service universe [1].
Although you'd think the simplest setup would be to merely configure for the subdomain case (https://www.example.com), in my experience using the apex domain (https://example.com) instead resulted in fewer complications.
So here's my recipe for using a custom domain with GitHub pages where Namecheap is the DNS provider:
