google search console – must the homepage be configured as canonical?

In the last 30 days, I have experienced real SEO and CTR boost on Google Search, only with canonical page configuration.

The new Google Search Console when indexing pages also retrieves information whether the page is canonical (set by the user) or not. If this is not the case, Google automatically creates a canonical page (which can sometimes cause Google to assign a false page to canonical).

Now, I have this challenge, my root page ranks very high and all other versions (/, index.php, index.html) are redirected to the root BUT when I let the root page be indexed by the console search, they do not show any information on canical page as it is not set up.

Now, organizing pages as canonical surely helps Google to rank the pages, but I'm confused about the canonized homepage. should I do it or not. (or am I just going over everything so that I lose all meaning on the canonical pages)

Thank you for any information!