I am using macOS high sierra version 10.13.6, if this helps
Don't know why you weren't asked this from the start.
Besides Preview, there is an app called Grab (which is replaced by the new Screenshots app in Mojave and later). However, it doesn't give you access to your current Screenshot save location.
Try looking for your screenshots in the Recents view in Finder. Accessible via the Go menu in Finder if you do not have it in your sidebar.
If they are in Recents, you can discover what path they are saving to. You can set the default back to Desktop if you want. Just google "change where screenshots save High Sierra" to find the instructions for doing so via Terminal.
If they are not in Recents, then you can still try setting the default back to Desktop in terminal anyway to see if that fixes the problem.
Otherwise, they aren't being created at all. Probably a 3rd Party app is interfering with the keyboard shortcut. It certainly seems like the icons in your menu bar in the video are doing something related to your keyboard input. Isolate by going to System Preferences > Users > Your User > Login Items and removing everything, then log out of your user and back in, to see if the issue persists.
If it persists, create a new user in the same System Preferences menu and log into the new user to test there.
If it persists in a new user, try uninstalling your 3rd party apps one by one. The culprit is likely a smaller app that may have special instructions for uninstalling (rather than just trashing the app in Finder) to make sure no helper processes are left behind, so consult with the instructions for your specific apps. It may be an app you have already trashed, but that required special uninstall instructions and is still partially running.