I have not read every single post, but to answer your first question directly - it is not known. We have no evidence to suggest why this is helpful, it's just the case that SIDS occurs more frequently to babies under six months when they are sleeping alone than when they are in a room with you.
I just kept DS downstairs with me.
I am surprised at that NHS advice that Buglife posted. I've seen it before, but it surprises me because NHS advice is supposed to be evidence based, and as far as I am aware, the "quiet dark room" and the idea of "teaching a baby day from night" is entirely anecdotal and merely common practice, rather than evidence based. It even says "It's a good idea to..." - which is odd to me. I'm tempted to question it. Especially the direct instruction to avoid feeding or rocking a baby to sleep - fine, perfectly okay strategy to go with, but it's not evidence based. And it's not the only way.
This isn't about advice that I agree/disagree with, BTW. I'm perfectly supportive of advice which states that it's safest for babies to sleep in cots, even though I'm a supporter of co-sleeping, because the evidence that we have available DOES show that cots are statistically safer. That makes sense. But I've never seen any evidence at all (aside from anecdotal, which doesn't really count) that it's important to teach babies day from night or that the method suggested achieves this.
I think that you must do whatever is practical and safest. It was totally practical for me to have DS sleeping downstairs with me for daytime naps. I suppose that I sometimes left the room, but I was in and out, pottering, and I had a small house. It wasn't like I was going a long way away from him. Conversely, I would imagine that if your baby is very restless and easily disturbed, it would be better for them to be in a quiet and dark place to sleep. And then it would be madness to expect parents to sit quietly in with them. But it's just saying that it's safer for them to be with you, they don't know why, so if it's reasonable to do that, then do it.