I maintain clients' Wordpress websites (~25 installations) as a part of my job. To do this quickly and efficiently I use Jetpack's "Manage" feature. In case something went wrong I have fully automated backups (Updraft plugin).
It works very well, but every time I do an update (which is a lot with Wordpress, themes and plugins update frequency) I have to go to the website and check if everything is ok.
Over a time I have experienced problems many times - mainly simple problems like "unable to connect" error in some plugin, not loading plugin in a sidebar, a changed font size after theme update and so on. Not very great problems and almost always easy to fix, but sometimes I overlooked them and made my client call me that his website is broken.
I wonder if there is any "proper" way to make post-update checks. I can't imagine that big agencies do it manually after every update for every client. The best idea i came up with is to automatically make screenshots of strategic places after every update and than watch them manually. I haven't found any articles about this in the internet, so I'll be very grateful if you can give me some clue how to do that.