Nothing like consistency in advice...not! I've heard these things about Tresillian staff advice
I have no idea what the notion of putting the baby to sleep whilst awake is based on. It has no basis in research and really, do you go to bed whilst you are still wide awake? Do you go to sleep easily if you are wide awake? I know for my own part, I need to do something that sends me off to sleep first - wear myself out with reading or housework (don't recommend the latter!), or sex...funnily enough, this activity is closely related to BFing, as it involves the same hormones - the ones that promote love and...sleep! This is why cuddling your baby in a sling sends baby off to sleep, it's a close comfort activity that promotes sleep and security.
I do hear you about your baby getting heavy in the sling, and this makes it harder. Have you tried sleeping your baby in your bed, by lying down beside him when you take him out of your arms/sling? At two months, there's not much likelihood of him rolling off if he's in the middle of the bed, I think I recall (!!).
My DS didn't sleep for too long, either, for months on end, and he got easier and easier to transfer out of the sling and into the bassinett, then to the bed. We didn't have the sling from day one, we got that at about 4 weeks, so there was an adjustment period there again, and the HAB became our best friend and sanity saver
ETA: I wasn't shy with the dummy, either - DS still uses one at night and for daysleeps cos he probably feels like he's still on the boob that way!