Stop screen recordings saving to desktop on MacBook Air

When you said you changed the Save Location, was that done in Options after the Shift-Command-5 to start a Screen Recording. I only ask because when I change the Save Location there, it persists with the next recording and nothing is saved to the Desktop if have chosen a different folder such as Documents.

How to record the screen on Mac - Apple Support

Or are you using QuickTime Player for a Screen Recording? That way always defaults to QuickTime Player to open the recording after it is stopped and the location of the recording is on the Desktop and remains there even if you choose to also save it in another location using the File menu in QuickTime. That workflow has been the same even on Sequoia when I tested on an older machine, and not specific to any Tahoe point update.

to prevent the recording "save to" location reverting to "desktop" after each recording, have you selected "remember last selection" in the recording options? when selected, it will use the same location for each recording.

and if I may add my two cents regarding the "save to" location: I have, in the past, encountered issues with longer recordings stopping / not saving unexpectedly whenever I have selected anything other than "desktop" as my "save to" location. this has been especially true when "save to" is set to an external drive. so I now use "desktop" as my "save to" every time and afterwards I move them to my desired location manually. I haven't had an issue since.

The big questions, I think, are HOW you're initiating the recording, and what you actually want to happen.

As already noted, the Screenshot.app Options lets to choose where to save, and whether it not it should remember that setting, so that should take care of that, however, you also say:

"... Is there ANY WAY to stop automatically saving any "Screen Recording" I make"

If you set the destination to "Clipboard', it's saved on the clipboard rather than to a file, so that saves you having to clutter your desktop/documents/wheverever with screenshot files. You can use that to paste the screenshot into whatever application you prefer, rather than saving to disk as a discrete file.

If you using the keyboard commands to invoke a screenshot (Command-Shift-3 for full-screen, or Command-Shift-4 for an area), you can use the Control key to toggle copying to the clipboard rather than saving to file (e.g. Command-Shift-4 -> Screen Capture to disk; Command-Control-Shift-4 -> Screen Capture to clipboard)