How right is defining elements with the number of protons?
Most of chemistry doesn't care a whit about all of the properties you named: they're in the realm of nuclear physics, as noted by user badjohn.  Even radioactive decay is something of a black sheep, included in physical chemistry as a convenient example of a first-order kinetic process. To my knowledge, it's not substantially used in conjunction with any other sub-discipline of chemistry, as it'd be more of an annoyance than anything ("Gah, the cobalt center of my complex keeps changing into nickel!").  It does of course have plenty of applications (medical imaging, nuclear power, etc.), but they're arguably in the realm of physics, with any attendant chemistry being a secondary consideration.

Magnetic moment, neutron count (isotopes), and nuclear quantum states do have some slight influence over certain spectroscopic properties (primarily NMR, as far as I know), but other than the somewhat special cases of the spin isomers of hydrogen and the kinetic isotope effect of deuterium vs. protium I'm not aware of any appreciable effect of these on chemical reactivity.

For the chemist, the identity of a nucleus is sufficiently defined (within an experimentally measurable perturbation, at least) in virtually all cases by its atomic number.