Visitors see "Briefly unavailable for scheduled maintenance" during updates

That is WordPress's own message. While WordPress updates itself, or updates plugins and themes that are in use, it puts the site into maintenance for a minute, before any plugin has loaded, so this plugin's page cannot be drawn then. What the plugin can do is leave a copy of your page where WordPress looks at that moment. It is off until you switch on Settings → WordPress Updates → Show your page while WordPress installs updates. Visitors then see your design and logo with a headline and message of your own, and search engines get a 503 with a Retry-After, so they come back later.

The copy has no sign-up form, countdown or password link, because nothing on the site can answer them mid-update.

On a network, only a network administrator can switch it on, because the one file serves every site of the install. Importing a settings file never switches it on or off either; do that on the Settings screen.