I have never found search console useful for anything other than random 404's and one-off server errors.
When a site is hit by an algorithmic penalty there is never any clue as to why.
This is my experience as well. If someone could mention how to examine the console to improve a site after an algo change, I would absolutely love to hear it .
I also find it mystifying and I’m not a SEO expert, but one way I use it is to compare ranking positions over time for specific pages and queries. That will help you identify where exactly the traffic came from and where you lost it. A lot of times this has to do with gaining or losing the #1 spot to a competitor.
If you don’t see a ranking change, it could also indicate seasonality in your visitors or external events. If your website is brand new, this could be hard to detect otherwise. For example, recently I saw a traffic spike for a random blog article, and search console help me see that people were searching for that topic because Elon Musk tweeted about it the day before.
Another helpful feature is to inspect particular URLs and make sure they are indexed—sometimes if you have multiple similar pages and set the canonical URL incorrectly, Google will try to de-dupe the results.
