IIRC the injection will be either syntocinon (the same drug as is used during induction) in isolation or it will be a combination of synotcinon and ergometrine - often known as syntometrine. The ergometrine is the component that seems to get the blame for many of the "minor" side effects (such as nausea) and is also what encourages the cervix to close again, starting the clock ticking re. placental removal. As far as I know the ergometrine is included as the combination is more effective at stopping excess bleeding from the uterus.
Talking to MWs who have perhaps more than average experience of physiological 3rd stage, it seems that overall blood loss will tend to be similar regardless of whether the 3rd stage is managed or not - just that if the injection is given the immediate blood loss will be reduced but the lochia will then be heavier and last longer.
IMO whether or not to have a managed 3rd stage is a very personal decision - and is one that should be flexible depending on the way the birth has gone to that point and how the 3rd stage is progressing. A physiological 3rd stage is dependent on natural oxytocin being produced - and the calmer and more relaxed the birth has been the more oxytocin is likely to be produced (and can be increased by skin to skin with new baby, starting BF, peace and quiet, etc). The more medicalised the birth has needed to be the less likely a physiological 3rd stage is to be appropriate or safe.
Also IMO, the biggest risk in a physiological 3rd stage is having a MW who is unfamiliar with the practice and tries to make it fit into the paramaters and guidelines of a managed 3rd stage. I've heard of MWs applying significant cord traction (which should never be done with a physiological 3rd stage) and withholding the baby until the placenta is produced (not likely to bring about the desired result). If you are planning a physiological 3rd stage it is prudent to make sure that both you and your BP are very well briefed on how you expect things to progress, and possibly having it written up as well such that it can be handed over if the situation seems to require it.
Its worth remembering that (contrary to what some women seem to be told) you can have the injection at any point - immediately after the birth, after 5 minutes, after a fixed period of time if things aren't progressing, after an indefinite period of time if you're getting bored of waiting, not to mention if your medical condition seems to dictate it might be a good idea. Its not something that needs to be cast in stone before you even go into labour.
Previous PPH is something that should be considered - but if you're interested in a physiological 3rd stage I'd always suggest getting hold of the hospital notes from the previous labour to try and understand whether there's any underlying reason why the PPH occurred. For example, I had a PPH with DD1 - but it came from a vaginal tear rather than the uterus so no-one I've talked to has suggested that it is a reason to avoid a physiological 3rd stage in any future births. There are factors that can increase the risk of PPH (such as assisted delivery) - if a lot of those factors were present in a birth that resulted in a PPH, does that mean that the individual woman is prone to PPH or that the way the birth progressed and managed predisposed her to a PPH in that birth?
As to "why wouldn't you have it" - its actually quite scary how little research was done on it before its introduction in the 70's. But it became standard practice such that everyone seems afraid not to have it. I'd also re-frame the question and consider why, in the absence of clear medical need, why would you have it?