No, it will not automatically/easily do that.
Google has not made that an option for every company to natively implement it yet.
I actually got my original doorbell to do this many years ago using Tasker in Android and some workarounds, but it was complex to set up.
Google just last week released a scripting feature for Google Home to create custom automations using YAML code. I haven’t played around with it much yet, but it’s very possible that Google will allow setting up doorbells to push a Livestream to a Google display through that now.
In general though, the answer is that this is still beyond the capabilities of the average user to set up, but Google is starting to add more options to Google home this year, so it’s looking more and more likely to be easier to do soon.