In my experience, there is no way to initiate a video recording or a snapshot from a Google Nest camera without retrieving it from the camera's history (without using a 3rd party screen capture tool of some kind).
For the older Google Nest cameras and doorbells that work in the Google Nest app, you can edit and save clips from the camera's 24/7 Nest Aware history (if you have Nest Aware Plus), or you can save clips of detected events, using either the Google Nest app or the home.nest.com website. I have seen various posts from customers reporting limitations on how long a clip can be, but I don't have any first-hand experience with that.
For the newer Google Nest cameras and doorbells that work ONLY in the Google Home app, you can only save clips of events that have been detected by Google Nest and saved as events in the Nest Aware event video history, though you can still view 24/7 history (if you have Nest Aware Plus).
Thank you for sharing your Feature Request with us. I can understand your frustration with not being able to manually create clips of footage if there was no triggered event. We’re always looking for ways to improve, and appreciate your feedback. You can send it at any time using your devices by saying, "Hey Google, send feedback," or by following the steps found here. This thread will remain open for 24 hours for any follow-up questions or concerns.