There was an old theory about SIDS which was that it was related to neonatal sleep apnoea, which means pauses in breathing. This is the reason that the home monitors were first invented. This has been disproven but the companies conveniently fail to explain this part, allowing the misconception that they might help in some way to pervail, because it provides a useful marketing angle for them.
In reality neonatal sleep apnoea is normal and is not thought to be associated with SIDS. Newborns will pause in breathing, which can cause these monitors to alarm. Parents then have the perception that the machine has "saved" their child but realistically nothing has happened. Or in a way, it's similar to the way that dopplers work - the doppler can show you that your unborn baby still has a heartbeat, they can't realistically prevent miscarriage/stillbirth or reassure you that it will not happen, they can only act as reassurance that it has not happened yet.
I do think the ones which measure oxygen saturation levels have a different mechanism and potentially these could alert to a baby who has got themselves into a potentially hazardous position in time for a parent to intervene. However all the oxygen saturation ones are very expensive.
With certain kinds of anxious feelings, it can actually be counterproductive to rely on this kind of reassurance as looking for the reassurance causes you to seek it more etc and it can increase the anxiety and fear. In extreme cases, this is the mechanism behind OCD. But postpartum you are more susceptible to this kind of thought loop, so particularly if you have history of any kind of anxiety, OCD or other mental health difficulties, I would be really cautious about anything which rewards reassurance-seeking in this way.
Then the other issue with them IMO (and has been pointed out by experts, I didn't make this up) is that sometimes people can use the reassurance they get from these machines in order to put their baby into a more risky sleep situation than they would otherwise feel comfortable with. This is a problem because the risk is not actually mitigated by the machine. This is one of the criticisms of them. IMO, either take the risk knowingly (e.g. I chose to co-sleep with my babies, avoiding alcohol/smoking and keeping bedding away from them; I also chose to let them nap in inclined containers e.g. car seat, swing, bouncer, when I was in the room and awake.) or do not take the risk. Don't let a monitor change what you would have done otherwise.
If you are struggling with anxiety/intrusive thoughts around safe sleep speak to your midwife as they can refer to the perinatal MH team.
Breathing monitors have definitely been around for more than 15 years as I remember them with my eldest (Angelcare was the brand then). The Lullaby Trust or FSID as they were called then used to loan them out to provide reassurance for parents after a SIDS death. I am not sure whether this was stopped because we now know that reassurance-seeking in this way can sometimes be self-perpetuating.
I think the modern way they are marketed can be really predatory, and I find the companies and marketing shady for that reason. Personally, I would actually avoid giving money to these companies if possible. We used audio monitors and video monitors and found them useful to see if the toddler was asleep or not (we didn't really need them for babies as didn't leave them alone to sleep) but I would have avoided Angelcare, even though they also make video/audio only monitors without breathing monitoring, because I don't want to give money to companies which prey on new parents' fears like this.
So my conclusion seems to be - I don't know whether it does anything, but as a dilligent middle class parent, I absolutely do need one.
IMO this just says that you've been sucked in by the marketing. I do not think you need a breathing monitor to be diligent, regardless of how much money you have. Being diligent IMO is things like making sure your cot meets the right safety standard and is used correctly, being aware of safer sleep advice and following as much of it as is practical, making sure your baby gets the right health check ups, vaccinations etc and following advice from HCPs.