When a site is hit by a Google update, how do you find the reason for the hit?

What all analysis do you conduct?

Google console??

Backlinks, keyword anchor ratio, content etc

You will receive the message if your site have a manual penalty.

Otherwise, you have to check all the other possible reasons.
The most common issue for the google hit will be backlinks. Check the did you lose any backlinks or did you put any new links. Over optimized content will make issues as well. Make sure you don't have any copyright issues. You may be outranked by your competitors as well.

Check your rank tracker and see what sites overtook you and how they differ from your site. This involves all areas such as on-page, speed, UX, backlinks etc etc

Then read up on what other research people have done and apply it to your industry and niche to see if it all adds up.