Question

why the properties of the elements are the periodic functions of their atomic numbers and not the atomic masses?

Answer 1

If atomic weight or mass is used to arrange the elements in the periodic table then the isotopes having the different mass need to be placed in a separate position.
Isotopes have similar properties and should be placed together.  This is a major drawback of  Mendeleev's table.  Isotopes have the same atomic number and using atomic number as the criteria for arranging the elements puts isotopes in the same place in the table. This is a better way for arranging the elements in the periodic table and was adopted as the basis of the periodic law.