How to remove a Recent Item in the Windows 10 Search Box
Solution 1:
The X to delete an item is still there (Screen shot below). You may need to click or hover on the item to see the X. This is evident in the screen shot. Also try restarting to check
If for sure the X is not there, try:
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Updating the V21H1 (may be an optional update - check and see).
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Run DISM and SFC
(1) Open cmd.exe with Run as Administrator.
(2) DISM.exe /Online /Cleanup-image /StartComponentCleanup
(3) DISM.exe /Online /Cleanup-Image /Restorehealth
(4) SFC /SCANNOW
(5) Restart when all the above is complete and test.
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Then make a new test Windows User Account (User Profile), log in, try search a couple of times and see if the search history can be deleted.
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If none of the above fixes the issue, then try a Windows 10 Repair Install from the Microsoft Media Creation Link. If the user account is OK, use the Repair option to Keep Everything. That way, nothing is lost.