Because abortion is a constitutional right in the US, and it would immediately be enjoined as a flagrant violation of constitutional rights (at least pending the current court's likely, but not certain, overruling of Roe v Wade and Casey).

Instead, this is a way to get around constitutionality since it's not the government who is violating the constitution. Never mind all of other constitutional bits about due process that are being horribly trampled on (such as allowing people to sue without standing...).