The "usual argument" does make sense to a certain extent, for it is the legal rationale for why the court's decision in Roe v. Wade didn't even meet the most basic requirements for legitimacy. The protection of the unborn is a valid application of the Due Process Clause and the 14th Amendment, and would probably have to be pursued under separate court actions. In other words, Step 1 has to come before Step 2.
I do agree with your general point, though. The notion that an innocent person could have his/her right to life legally enshrined under the laws of one state but could be legally killed under the laws of another state makes no sense at all.
Did the Founders intend to protect unborn children as "persons" who would be subject to Constitional protections? That would have to be answered before you could make that argument.
On the contrary -- that's the entire point of federalism. The people of each state are free to make such choices via the legislative process. It makes just as much sense for that to be decided at the state level as it does at the national level. You have no problem with each country deciding such things as sovereign actors and you should have no problem with each state doing so under the founding father's concept of federalism wherein each state is as free as an independent nation would be in social matters like abortion where the Congress is given no power to legislate.