I can't get SUMIF to work in all cases. The following refers only to criteria argument.
works: 1 "1" ">1"
also works: K3 where cell K3 contains either numeric 1 or string '1

does not work: K3 when cell K3 contains ">1" or "*"

How can I make the criteria be another cell which contains string values and/or an operator?

The inherent data in list being compared is numeric.