Does anyone know the (specific) reasons that it is recommended that old LTS server users wait until the point release to upgrade? Is it just a matter of bug fixes and waiting for improved stability for production servers, or is there something more fundamental than that?
Yes, most users that use LTS are conservative, even more so on the server side, so we don't enable prompting updates until the point release.
It is correct that there will be bugfixes published between now and then, and that 16.04.1 will be a culmination of all those bugfixes.
If you're itching to move to 16.04 right away though, you can follow these instructions to move to 16.04 immediately:
You can see milestone information and status of specific bugs for 12.04 here: https://launchpad.net/ubuntu/+milestone/ubuntu-12.04
and 12.04.1 here: https://launchpad.net/ubuntu/+milestone/ubuntu-12.04.1
Depending on your use case, these lists may provide a more complete picture for you regarding whether or not you should wait for the point release or not.