The MW in that case should not have pulled on the cord to get the placenta out.
Cord traction is used in a MANAGED third stage after the synto injection has been given. NOT in a natural 3rd stage. In a natural 3rd stage, the MW needs to keep her hands off until the placenta is out. Cord traction, especially if she's rough, can cause the placenta to break.
After the placenta is out, it must be thoroughly inspected to ensure all the parts are intact, and none have been retained. There are two lobes to a placenta, the cord, and the membranes. The MW should inspect that the membranes have all been passed. She should perform this inspection in front of you and show you your intact placenta. She should monitor you closely for further blood loss and ensure it is of a minimal/safe amount.
It is worth mentioning that you could have a natural delivery of the placenta, delayed cord clamping, AND THEN have the synto injection after the MW is satisfied the placenta has been entirely delivered. The synto would aid your uterus in shrinking and could provide the reassurance you need that your blood loss will be minimal.
Also worth noting: in either a managed or natural 3rd stage it is VITAL that the MW is competent enough to check and ensure that all of the placenta has been passed.
That story is tragic, but it was negligence, not the fault of a natural 3rd stage per se.