I've put some code onto the tmtest site that will display a message at the top of the page during the critical phase of the site update process when the DB is being repopulated. I decided to let subsequent errors continue to happen, since some functionality might work during this phase.
It's achieved by a file being created right before the mysql command executes that will update the DB, which is then removed when the command completes and the DB has been recreated. I'm manually put that file into place on the tmtest site for now so everyone can see the message (red banner) that will show up.
I'm planning to include this functionality when I run the next site update, likely tonight.