No, Google gets a cut for purchases through the Google Play Store.

Google does not take a cut if you use an alternative app store (which isn't possible on iOS)

I assume the 30% platform fee is the reason why nobody uses Google Play and everyone uses other app/game stores instead.

Everyone uses Google Play because it's convenient. But as a notable example, Fortnite refused to use Google Play for a while precisely because of that 30% fee, and it worked out pretty well for them. Eventually they gave in and put Fortnite on Google Play. Although Google kicked them off later (they pushed an update which allowed users to bypass Google's 30% cut using their own payment system) so it's back to direct download from the website.

It worked out pretty well but they gave in. So why do you think it worked out pretty well if they came crawling back?

Which no one does and when even Epic tried it it was an abysmal failure.

Android allows you to update apps from outside the play store.

And this is why Google and Apple form a duopoly in the mobile app distribution and mobile app payments markets.