I would say find out which drug they use in that birth centre to deliver the placenta before making up your mind.
Syntocinon is the most commonly used, IM injection, works within 2-4mins, stays in the system around 40-60mins. Most common side effects - nausea, vomitting, headaches,
Some units use syntometerine (more effective for stopping blood loss post birth), still IM injection, generally worse side effects - nausea, vomitting, raised BP, palpitations, arrythmia, headaches, dizzyness
Length of 3rd stage with injection is usually within 5-10mins. Naturally can be over an hour (but often about 20-30mins).
Blood loss post birth is less with the injection - however blood loss in the weeks after the birth tends to be more. With a natural 3rd stage blood loss is heavier after the birth but tends to be less in the following weeks.
I guess a disadvantage of natural 3rd stage would be if you lose over 500mls they class it as a PPH which means you may be considered 'high risk' next time. However it might be worthwhile doing some research if you plan on a physiological 3rd stage as cardiac output is increased in the last trimester, so you have up to I think it's 40% more blood pumping around ready to compensate for blood loss post birth.
Also with a managed 3rd stage you don't have to 'do' anything but with a physiological you may have to use the help of gravity by squatting etc to help the placenta out.
Oh and the other thing about the injections - it isn't common but it does happen sometimes where the drug works really quickly and the uterus clamps down before the placenta is out, meaning you will have to have it manually removed.
Just make sure whatever you decide you read up on it all first and the midwives know what you want x