Okay, so bear with me if you would. I think I’m just being slow today.
I apologise in advance if the below analogous story fails on every level, but life’s easier for me to understand life when it’s explained through analogies.
Say Paul Peterson (PP) has a part-time job. It doesn’t really earn him enough to cover all of his monthly needs, and so his parents subsidise him with an allowance. Their only stipulation is that he not spend any of the money that they give him on drinks, drugs or cigarettes (abortion); he should only be spending the money on necessities that he would otherwise have to pay for out of his own earnings – clothes, lunch, social activities, etc.
How are Paul’s parents making sure that Paul is not buying cigarettes with his private earnings?
The analogy has a flaw. If Paul's parents don't want him to buy beer, they should make receiving an allowance conditional on not drinking. Paying for his non-beer related expenses is the way the parents acknowledge that he can drink beer provided he self funds his drinking.
Let's tweak the analogy a little. Let's say Paul wants to be an sculptor and a doctor, but his parents want him to focus on being doctor.
If Paul's parents are Philistines and don't want him to pursue art, they would just make a deal that they will support him as long as he quit's sculpting. Paul's options are A) Quit art or B) figure out how to make art full time for a living to support himself. He chooses option B and becomes a full time sculptor, and his parents stop talking to him.
Now let's say Paul's parents recognize that even if they don't fully appreciate it, his art still has value to his personal development and broader society. They recognize that he could be both a talented doctor as well as an artist. They make a deal: Paul will provide them with receipts for all his medical school related expenses, such as tuition, rent, groceries, etc. and his parents will reimburse him for those as long as he performs well in school. If he wants to make sculptures, he will have to figure out how to fund that on his own, presumably by selling his artwork to fund the next project.
The PP federal funding is similar. The fed doesn't cut them a blank check. PP performs approved services such as STI or cancer screenings for free or reduced cost to the patient, then submits a claim to Medicare for reimbursement. The abortions are funded through charity donations or by the patients. When you take the federal funding away, it only hurts the patients that go there for the Medicaid approved procedures. It doesn't really do anything to the abortion side of the organization. In fact, it may force the organization to focus solely on performing abortions for profit.