The key difference is share.google, as you mentioned, is for Google controlled properties whereas goo.gl allowed shortening any arbitrary user provided URL. Which opened up a giant can of worms with Google implicitly lending its brand credibility to any URL used by a scammer, phisher or attacker.

You can generate share.google links on chrome for any arbitrary url.

How? I just tried each of the Share options for this thread in the desktop Share menu, and they all used the full URL. Including the QR code which I verified by saving as a PNG and scanning it outside of any Google app. I also haven't found any Share option in the iOS app either that doesn't use the full URL. But harder to test on mobile given the various permutations of sharing between random apps.

Sorry, may be wrong about chrome, but you can definitely do it in the Google app on Android.

I just disabled the setting recently. Here's a link to a Reddit thread talking about the feature

https://share.google/GlkbFpGznyLszpoyc