Agreed, "My DNA is my personal property," is a weak argument.
"My DNA is my personally-identifying data," is a better one.
The question of what rights we have to our personally identifiable information is not yet answered fully, and the debate is an important one (e.g. consider GDPR and the "right to be forgotten").
Data's worth is recognized today by those who are harvesting it from us. The mountains of data which only a few multinational corporations can gather and store in globally-distributed data centers are imperative to machine learning and the artificial intelligences they are making, which will soon direct a great many of our daily conveniences at potentially great profit to them.
So it is important for us to today define what rights we demand to such a valuable resource, which is created solely by, and living within, the only item any of us has true natural possession of: our very bodies.