I have updated the docs on content storage now.
Technically if the migration failed or you closed the browser tab before it finished you could have a mix of content. Although in that case you would just run the migration again and it would continue after you addressed the issue.
I like the alert just because the browser will give you a notice on the tab that it has finished. We do plan on redesigning the Cornerstone settings dashboard in Cornerstone 7.7.0 so we might revisit more styled modals in that release.