The really sad thing is that they actually had this.
Hangouts on a Google Fi phone used to let you consolidate everything to hangouts - SMS, Chats, Calls, etc. It was a pain it was limited to Fi, but it worked just fine. You could also pop open Hangouts in Gmail in your browser and there all your sms messages were, nicely grouped and tracked.
Then apparently hangouts stopped being cool enough - so Google decided to shutter it in favor of "Google Messages". Technically - "Messages for web" also supports the same feature set as the integrated hangouts, but it FUCKING SUCKS.
The phone/browser syncing is bad, the UI is a complete regression. Just the process of enabling it required me to actually contact support at google fi (you have to open settings, disable previous sms settings for fi, stop hangouts, update your google account, and then opt into a beta).
Basically - Hangouts did exactly this, but Google continues to act like Google, and shutter any product that might actually be a good foundation to build/maintain, since Google has shown AGAIN AND AGAIN that it can't actually maintain any product.
Google can innovate just fine - it's the follow through that they lack.