OK. I decided that the omnimigrate made a mistake and did actually upgrade the database as it reported, but forgot to modify the version.txt files.
So on the server running 6.20, I changed the 6.00 string in all the version.txt files to 6.20 and then did an omnisetup.sh from the 7.0 media. It worked without any errors.
The GUI looked normal (reporting version 7.0 for all the components) so I applied the 7.03 patches and there were still no errors. But similar to the 6.00 version.txt problem, the GUI reports all of the components as 7.0, not 7.03. (big sigh...)
However, everything is working fine.
It would have been extremely helpful (and saved a lot of time) if there was a method to query the 'real' version of the database.