I'm not convinced that BTS had anything to do with the drop in SIDS. The BTS campaign occurred at a time in the US when smoking rates started falling off a cliff. Between 1990 and 2018 smoking has dropped 42% [1]. Between 1992 and 2017 the rate of SUID (includes SIDS and a handful of other things that used to be commonly lumped in with SIDS) has dropped... 42% [2].
Yes, correlation does not equal causation. That said, by definition SIDS has no known cause and smoking has been identified as one of the biggest risk factors.
Edit to add: one other thing that I think is hard to track. As a parent that tried to sleep two kids on their backs, they slept terribly. In the middle of the night, in sleep deprivation fueled desperation, the two options we saw were to put them on their belly or bring them into bed with us. Cosleeping is a far greater risk factor of SUID than belly sleeping but people make that choice every day because of the shaming that doctors and nurses put on belly sleeping.
[1]: https://www.forbes.com/sites/niallmccarthy/2018/07/26/poll-u...
[2]: https://www.cdc.gov/sids/data.htm