The newer Google Nest cameras and doorbells released starting in the fall of 2021 work ONLY in the Google Home app and NOT in the Google Nest app or on the home.nest.com website.
Note that, with the Google Nest Doorbell (Battery) you only get event video history even if the doorbell is wired, because it always runs on battery power. If you have a Nest Aware subscription, you get 30 days of event video history; otherwise you only get event video history for events in the last 3 hours.
Here is Google Nest's help topic on viewing "History" in the Google Home app.
Thank you for your feedback on this issue, I fully understand the frustrations of not having your new Nest cams in the Nest app or available on https://goo.gle/3A7x7ug web app.
However, the new Nest cameras and doorbell are exclusive to the home app because we wanted to create an integrated experience with your speakers and displays. A nest aware subscription applies to all of your cameras, speakers, and displays in the same home structure. Thank you for your continued patience, and support.