For now, I've found the following workaround, should you decide it is worth the trouble to have a default webapp:
1. Leaving the context-root setting in the project as is (i.e. don't try to set it to "/"), deploy the project to the Tomcat server.
2. Double-click on the Tomcat server in the Servers view to open the configuration editor.
3. Switch to the Modules page of the editor and select the web module for the project.
4. Click the "Edit..." button, clear the Path field, and click OK.
5. Save the Tomcat configuration and close the editor.
6. Right-click on the project in a navigator or a resource in the web module and select Run As -> Run on Server.
7. Select the server to which the project was previously added and click Next.
8. Click Next again. Uncheck the "Update context root for Web module..." checkbox and click Finish. This will add a setting that prevents the change made in step 4 from being overwritten by the project's context-root setting. The browser should appear with the correct default webapp appropriate URL.