Why aren't column operations used to reduce a matrix in its echelon form?

At a fundamental level, matrices are objects containing the coefficients of different variables in a set of linear expressions. Each row is for a single expression, and each column is for a single variable. When solving sets of equations, we can combine equations by adding or subtracting the equations, or multiplying them by a factor; it wouldn't make sense to multiply the coefficients of a single variable in all the equations by a number, or subtract the coefficient of one variable from that of another variable in all the equations. Hence, we perform operations on rows (coefficients in expressions), not on columns (coefficients of variables).