EllisLab Support helped me figure this one out, but just in case anyone else ever runs into this problem, here was the solution:
It was presumably a third-party add-on that was responsible for the error, and the add-on update process is the next-to-last step the update wizard runs before updating the EE version number in the config.php file.
I just needed to check that the EE 2.7.3 changes were reflected in the database, which meant checking that the following database fields were updated to a length of 75 characters.
- exp_members.email field
These had, in fact, been updated correctly, so all I needed to do was update the version in the config file to "273," remove the "installer" directory from the server, and log into the control panel to update the add-ons.
Now everything works just fine. Many thanks to Robin Sowell at EllisLab Support for helping me sort this out!