It isn't typical of abortions. Nor is it typical of clinics where abortion services are provided (AFAIK there is no such thing as an "abortion clinic," which would imply a clinic that ONLY does abortions).
Facts on Induced Abortion in the United States
Scroll down to the pie chart showing when abortions are performed. 88% of abortions are performed in the first trimester (first 12 weeks) of pregnancy. Only 1.5% of abortions are performed after 21 weeks. The third trimester begins after 24 weeks, so that would be an even smaller percentage. The number of abortions performed using the procedure you described is vanishingly small.
Under the guidelines of the
Roe decision, states are permitted to ban abortion in the third trimester (or after viability) except to protect the life or health of the mother. Most states do in fact impose severe restrictions on late-term abortion.
So the Court and most state governments recognize a distinction between late and early abortion. So does common sense. A fetus not long before birth shares many characteristics with a newborn baby, and in most cases could become a newborn baby without further gestation, at worst requiring treatment for the hazards of premature birth. Such a fetus, if outside the womb, would be regarded by common sense and ordinary language as a "baby." This is not true of an embryo in the early stages of pregnancy; whatever logical basis you might argue for treating them the same (ultimately based on religious arguments inevitably), these arguments must overcome the common-sense instinctive reaction which does not treat them the same.
What you are doing here is taking the natural revulsion people have over the killing of a newborn baby and trying to transfer that natural revulsion to an object for which it is not natural at all.
At most, you are presenting a valid argument here for restraints imposed on late-term abortion, which in most states already exist.
To use this as an argument for restricting or outlawing abortion performed in the first trimester is not rational.