Nick wrote...
Enter the criteria <"" in a cell then reference the cell instead of
typing
the criteria directly in the formula

B1 contains <""

=SUMIF(A1:C1,B1,A2:C2)

First off, B1 would be contained in A1:C1, so bad choice of cell to
contain the criteria. Then, that criteria doesn't do what you think it
does - it'll exclude cells evaluating to the string containing two
double quotes. To exclude blank cells, use

=SUMIF(A1:C1,"<",A2:C2)

Note that this won't exclude columns in which the row 1 cell evaluates
to a zero length string, "". To exclude those as well, easier to use

=SUMPRODUCT(--(A1:C1<""),A2:C2)