Afaik, babies have weak neck muscles and can't lift their heads. I remember also reading that really small babies can't breathe through their mouth. So it seems easy to imagine a situation where the babies airway is blocked and it can't help itself.
Basically the SIDS risk as understood currently is like you said: obstruction getting in the way, baby not waking...bad outcomes. Co-sleeping with the parents increase risk by bringing pillows, blankets, themselves into the bed to add obstacles or opportunities for the infant to suffocate.
It's a highly nuanced topic. In my parent's day it was common for babies to be put to sleep on their stomach whereas now the advise is to back sleep even though there's increased risk for flat head, but lower risk for SIDS. Once the kid is able to turn themselves it becomes much more simple as they'll move around throughout the night so back/stomach becomes a non-issue once they reach that age.