@ SH
“Didn’t have ANY? Really? Have you ever looked inside a Rambam? He didn’t have ALL, but he definitely had some. Such as הוגה בתורה ועוסק במצות כדוד אביו.”
That first criteria taken on it’s own would fit many Gedolim. Using your logic I could say that the head of the secular Israeli government fulfills one of the criteria by waging war against the enemies of the Jewish People. Those criteria are parts of a whole.