If atoms are made of smaller parts such as electrons

Why are atoms considered the smallest particles of an element?

Atoms are not considered the smallest particles of an element in the field of physics. They are however the smallest chemical component of an element, as they are the smallest part of a molecule that has discernible chemical qualities that are currently understood or measurable.

Once you change the atom, you no longer have the same element.

Remove one electron and you have an ion.
Remove one proton and you have a different element.

e.g.; Remove an electron from a magnesium atom and you have a magnesium ion.
Remove a proton from a magnesium atom and you have a sodium atom.

The reason is because the atom is the smallest particle that is identical to all the others.