Bah!
I figured out what was wrong, but I have no idea _why_.
I did a ps -A on the malfunctioning machine, and on another SME box, and compared the lists. Sure enough, there was a process missing on the failed system. The cure:
[root@foo /root] service httpd-admin restart
Shutting down http-admin: [ FAILED ]
Starting httpd-admin: [ OK ]
And life is good...
Dan