Quicktime does not save screen recordings
My set up:
MacBook Pro (16-inch 2019) running macOS Monterey 12.6.2
QuickTime Pro
I found a way around, get the autosaved file: Go/Go Folder/ and then type ~Library/ScreenRecordings
Then rescue your file from there. Not the most elegant solution, but it works. I actually placed that folder in my desktop for convenience.
Alternatively, if you check "Show Floating Thumbnail" from the QuickTime options, and double click the thumbnail after the recording is done, you can: a) send it via AirDrop to a different device or b) send it to photos. You do that by clicking on the share icon on the top right corner of the video window.
All these ways worked for me, I hope it helps.
MacBook Pro (2017 – 2020)