Both things you say are correct. And as far as I've seen on most recent studies it is indeed related to some sort of deficiency in the signaling from the brain

Both things you say are correct. And as far as I've seen on most recent studies it is indeed related to some sort of deficiency in the signaling from the brain to "wake up" when oxygen is low, with suggestion that there's a genetic component to it (sorry no link to the study I saw). 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.