I added "login with google" to my website. Should I go to the news media to brag about how google is launching an integration with me?

I disagree that integration is commutative.

Often, we see a feature which is important to free use of a computer as a general-purpose tool locked behind an ever-changing and/or poorly documented API in a closed-source, centralized, de-facto-government-subsidized project.

The power dynamics of that situation are not symmetrical, so it does matter which project(s) are using which API(s) of the other(s).