Honest question, what's Google's angle with flutter? It's one thing to be develop a toolkit for platforms they have a vested interest in, it's an entirely different thing to create such a toolkit for all platforms, including their competition.
I'm not totally sure what you mean by "a toolkit for platforms they have a vested interest in". Google has a vested interest in basically all the platforms out there. They might create Android, but they certainly make a lot of apps for iOS. I'm guessing they'd like to simplify things using Flutter. Mac and Windows might not see a huge number of native Google apps, but that could change over time - especially if they have a way of making a web version and a desktop version out of the same codebase. Imagine a native macOS Gmail client using Flutter. The same code that could compile to JS for the web could compile down to native for macOS.
Having something be open source and widely adopted makes a language and ecosystem stronger. The Go authors noted that they open-sourced Go because it would become stronger as more people used it.
And I wouldn't underestimate the soft power that comes from being in charge of one of these things. VSCode, .NET Core, and TypeScript are all free, but it's given Microsoft a huge amount of soft power in the ecosystem and a lot of credibility and mind-share.
With a cross-platform UI kit, you can definitely end up with a lot of power. Maybe the experience on Android feels better and more native because your engineers are putting their effort into the Android UI and its patterns more. That's not to say that they'd purposefully make the iOS experience bad or anything, but maybe the iOS-style widgets don't get quite as much polish and maybe developers get encouraged or defaulted into shipping Android-style Material Design apps to iOS. Again, I'm not saying this is malicious, I'm just saying that Google might spend more time on the pieces that are important to them. Google's web docs for Flutter could easily include sections for "and this is how you hook it up to Google Cloud".
I think it's also important to recognize that while Android has more marketshare globally, Apple is generally more popular in the US and generally more popular with upper-income folks. Apps like Clubhouse often launch as iOS-only. 11 months after launch and the Hottest New Thing™ is still iOS-only. If Google can convince developers to use Flutter for the Next Big Thing™, it guarantees that it will be available for Android.
I think there's a lot of value for Google. Your teams can build better apps in less time cross-platform. You potentially become a leader in the way that Facebook has with React, Microsoft has with VSCode/.NET Core/TypeScript, etc. You guarantee first-class support for your devices. You guarantee that the new things you want supported will get supported since you can prioritize your engineers on it - and potentially even get supported in the way you want. And potentially some nice tie-ins to Google Cloud and Firebase via first-class support, documentation, etc.