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#11
I don't think there's any account on W10 that can directly do everything. No matter you're a member of the administrators group, the administrator user or the system user... and this was the case with W2000 already, and probably WNT4 too.
Now back to your case, it's very possible that disabling UAC is the cause of the problem. The UAC-elevated environment is somehow "normalized", i.e. doesn't depend on the calling user for the most part. As an example, in the old pre-UAC times (before Vista), suppose you had two admin accounts Alice and Bob, creating a file in a directory such as Program Files wasn't different than creating one on your desktop, it would have a different security descriptor (i.e. owner and ACL) depending on the admin account used. Disabling UAC makes this happen again.
Anyway, you should have a look at the ACLs of 1:the file you're trying to remove and 2:its parent directory. Unlike it seems, the parent directory is likely not your desktop, but the public desktop : "C:\Users\Public\Desktop". You can check the ACLs either from the security tab in the file/directory properties, or from the command line, typing icacls <file/directory path>, e.g. icacls "C:\Users\Public\Desktop". The advantage with an icacls output is that it's easier to copy/paste here.
Knowing the owner of the file and its parent directory can be useful too, since the owner can modify the ACL of the owned file/directory at will, no matter what the explicit ACL is (except in one case, if the OWNER RIGHTS special entry denies that). You can get the owner from either the advanced button of the security tab, or from the command line with the dir /Q command.