The SIDS risk being parents rolling on top of and crushing / asphyxiating the baby. We just gloss over most of this sort of death by calling it SIDS to avoid having to charge the parents with a crime.
SIDS is totally different.
EDIT: Another important factor is where the sleeping occurs. A bed is very safe, a couch is not. She pointed out that an exhausted parent lying down with the child for an intentional nap is much better than them falling asleep sitting up on the couch.
A baby would not have been able to. Some people are hard sleepers, and even if you're usually not you probably still have days where you are.
Frankly, co-sleeping with a baby is irresponsibility of the highest form. It may have worked for the GP but for every one that it works, another 100 show up to my family member's office or to the morgue. There's no conspiracy in why part of the "how to handle a baby" guidelines they give new parents is "never, ever, EVER co-sleep". Its far safer to assume that the new parent is not in the less-risky population and will in fact eventually cause their child's death.
SIDS happens also when baby is alone in bed. You relating it to parents rolling on baby tell me you don't know much about the argument.