Its too late for me to try and dig out the data, but from memory a retained placenta can be more likely if you have a managed 3rd stage as, in the UK, they jab they use is syntometrine - which is a combination of syntocinon (encouraging the uterus to contract, which in turn encourages the placenta to detatch) and ergometrine (encouraging the cervix to close). Once the ergometrine has been administered you have a finite time window before the cervix closes - and if the placenta isn't out in that time then it will become retained.
The syntometrine can be given at any time after the birth - although it is often given in the UK as the shoulders are delivered, and it is often assumed that you'll have it. There is no reason why you cannot try a physiological 3rd stage and see how it goes and have the jab if/when you get fed up of waiting (be that after 5 minutes, 30 minutes or 90 minutes).
One reason often given as an advantage of a managed 3rd stage is that average blood loss is lower. However, I've seen some very experienced MWs express the opinion that initial blood loss may be lower but lochia tends to be heavier and last longer - such that total blood loss is probably comparable.
If you do decide you want a physiological 3rd stage bear in mind that, because in some areas its an unusual choice, its one that not all NHS MWs are very familiar with. As such its as well for you and your birth partner to be very clear on what you will / will not accept and be prepared to enforce your wishes. For example close contact with your baby is important, feeding if you can as the suckling action of the baby releases oxytocin (natural syntocinon) - but I've heard of MWs refusing to allow the mother to hold her new baby until the placenta is delivered. Also, in a physiological 3rd stage cord traction should be used extremely cautiously c/w a managed 3rd stage where its pretty standard and can be much more forceful.
As for is 5 retained placentas in the last 2 years a high number, in isolation its too small a number of cases to comment on. You'd really need to know what %age of vaginal births resulted in a retained placenta and how this compared with the national average before you could comment on whether this particular hospital was good or bad. I don't know where you'd get these data from.
Personally I'd aim for a hands-off birth with a physiological 3rd stage. If I had to have a managed 3rd stage I'd want the jab delayed until the cord had stopped pulsating (usually only a few minutes). Jab as baby born with instant cord clamping would be my least preferred option.
There's some more info here, with lots of birth stories of various different management strategies and some more links and references at the bottom.
Try not to worry - you've had one birth with no problems. Why should this one be any more difficult?