https://gerrit.wikimedia.org/r/c/mediawiki/extensions/OATHAuth/+/1186136 introduces some terrible hackery to display multiple buttons that each set a different value for the newModule field. HTMLForm really doesn't want to let you do this.
To make this cleaner, we should introduce an HTMLFormField subclass (initially in OATHAuth, we can upstream it to core later) that implements a group of buttons with the same name (similar to radio buttons, but with submit buttons instead).