It's not illogical to think that money you give an organization will benefit the activity that the same organization performs.
Your argument is not logical for a multitude of reasons.
1) No tax payer money is being given to grocery stores, taxi companies, construction companies.
2) these companies are separate organizations to Planned Parenthood.
I don't understand the argument why not to split PP into 2 separate entities: 1) a women's health clinic and an abortion clinic. If that were to happen, then the Women's health clinic could still refer women to the abortion clinics. But because the health clinic would be separate from the abortion clinic part of PP, the taxpayer money that is given to the women's health clinic wouldn't DIRECTLY benefit the abortion clinic.
Just split PP and this debate would be over. No one is saying in this current debate that the abortion clinics should close or that Roe v Wade should be repealed. Just stop funding organizations that conduct abortions. If you split PP then this debate is over.