SIDS is very poorly understood, but highest probability seems to be that it’s not germ related but related to breathing (more like complex sleep apnea for infants, where sometimes a baby has a messed up breath response. This tracks with SIDS seeming to have a strong genetic component). For what studies we do have, yes sleeping with the parent seems to have a marginal increase in SIDS, although less of one than things like having a loose blanket in a crib.
Ultimately risk falls off around four months though, so yeah maybe 2.5 months is a bit early to try weird things.
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.
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.
Ultimately risk falls off around four months though, so yeah maybe 2.5 months is a bit early to try weird things.