Thanks to JDRN, I at least understand half of what's going on now. Someone at Apple goofed -- if your computer is not already enrolled in the 10.15 beta, the enrollment tool that's available to download now won't work. Here's why.
If you go to beta.apple.com, login and then attempt to download the "Public Beta Access Utility", the link is pointing to this right now:
https://beta.apple.com/sp/downloads/projects/1001260/downloads/1010768
Clicking that link will download a DMG file. Double-click on the DMG file to mount the virtual volume. If you "get info" on the package inside this volume, you'll discover it was last modified on 05/30/18 -- yes, not 2019. If you install the package, it will set your Software Update catalog to this (I substituted
https:// with
XXXXX at the beginning because this editor keeps turning it into a hotlink if I leave it there):
XXXXXswscan.apple.com/content/catalogs/others/index-10.14beta-10.14-10.13-10.12-10.11-10.10-10.9-mountainlion-lion-snowleopard-leopard.merged-1.sucatalog.gz
Yes, this is the enrollment tool for the Mojave beta, not the Catalina beta. Oops! Want proof? Launch Terminal and execute this command to see where your Software Update is now pointed:
sudo defaults read /Library/Preferences/com.apple.SoftwareUpdate CatalogURL
You should see that "10.14beta" in the reply string.
Even though this installer is obsolete, you should still run it because it will create the SeedEnrollment file you need. Software Update will then check for updates and discover the "requested version of macOS is not available" since it's looking for the Mojave beta.
Once you receive that message, execute this command in Terminal to fix the the catalog location (again, where you see
XXXXX substitute
https:// because this board keeps trying to put a hotlink in the middle of my text!):
sudo /usr/sbin/softwareupdate --set-catalog XXXXXswscan.apple.com/content/catalogs/others/index-10.15beta-10.15-10.14-10.13-10.12-10.11-10.10-10.9-mountainlion-lion-snowleopard-leopard.merged-1.sucatalog.gz
Check for updates again and you should now see the Catalina beta is available.
HOWEVER: for me, at least, all this did was allow me to download the latest beta installer onto my MacBook Pro (15-inch, 2019). I tried using it to make a bootable USB installer disk so I could do a complete clean install -- booting on this USB drive still produces the error message about needing to update my installer disk. I can see it's attempting to download what it needs to update my disk but it fails halfway through.
I submitted a report through the Feedback Assistant for both of these issues. Of course, I'm just one of the horde who's sending feedback about the beta. Anyone with a bonafide connection to Apple may want to drop them a line that their beta enrollment tool has been incorrectly downgraded.
As for the other issue, I wish someone would figure that out so I can actually use the new computer I just bought.