"This means we manually log into the Dashboard, check for updates, view any notifications, make notes, and decide if any action needs to be taken."

We can confirm this but beside that we:
+ Check every page for missing or bad elements.
+ Keep the console open to see if any js error appeared.
+ Check the speed of the website.
+ Read Emails from the Security plugins to see if there is something strange happening. Many emails from the Security Plugin are just notes about updating plugins.

Let's talk now about plugins. There are many plugins with weekly updates, plugins like YoastSeo, WordFence, iThemes Security, but these plugins are just updating minor things. Taking this in count, 6 plugins to update it's not such a critical situation that's why we do this every Monday.

How Wordpress knows if there is any update to be made?
Wordpress is using a cron job to check the database for new updates.
When? Wordpress says: 7am and 7pm.

If we finish the updates at 6pm and you check the website at 8pm it's very
possible to find some plugins updates that were not made, but as we said
that's not a critical situation.