The Google Search Console system does send email notifications to webmasters to alert you to errors in your website such as mobile friendly issues, page speed problems, server issues, dead links and more.
Nowadays, it evens send you a monthly performance summary and if you have been naughty, perhaps a notification about a penalty!
If you connect your Google Search Console data to the Authoritas platform then you can create even better and extremely fine-grained SEO alerts based on changes in clicks or impressions in your GSC data.