I think I have figured out what I am missing. The domains cannot be managed from the main admin domain. There is a kind of chicken-and-egg situation here: the blogs for the alternate domains need to be created using sub-domains of the main admin domain first. Then you must log into those blogs (using the sub-domain for access). From there new domains are added (actually, just one domain, as multiple domains do not work out-of-the-box, but watch this space). From that point the sub-domain suddenly stops working and all links change to the alternate domain for that blog. Since the domain changes, you also need to log back in, as the browser needs new cookies.
If that process goes wrong, then there is no way to fix it from the main admin domain, and there really ought to be. Luckily the database table of domains is not too complex.
Like I say, watch this space, as I have a few patches to fix some of these issues.