Watched screen recording but it's not saved to the desktop

I recently recorded a session with my dad. I stopped the recording after the session ended (with the setting set to save to Desktop), clicked the icon that appeared on the lower right corner to open the file, and played back a few minutes to make sure it worked.

I then clicked Share to send it via AirDrop to my computer, which did not work; the status remained "cancelled" despite my efforts. I assumed I'd be able to access it through my phone by sending it through a third app, so I pressed cancel on the AirDrop pop-up window.

The entire screen recording closed and (apparently) vanished forever after the action. I'm extremely frustrated because I can't seem to locate this video.

This would be so much more useful if I could take a picture of taking a picture, but it seems Apple doesn't allow that.

If you go to System Preferences, Keyboard, Screenshots and note the key sequence for Screenshot and recording options, you can use that to record the monitor or portion of the screen, and specify where to save it. Options (which see) include saving to Desktop, Documents, Clipboard, Mail, Messages, Preview, and Other Location…. I suspect you saved your first recording to the clipboard, and the second to the Desktop, but I have no idea how you changed the destination between recordings.

Check ~/Library/ScreenRecordings/ (the Library folder in your home directory).