Unfortunately there are no guarantees, and as mentioned before, if there were some way to 100% guarantee a healthy baby we'd all sign up for it. You have to go with what you personally are comfortable with, and if you're fearful in a birth center, it probably isn't the place for you, and may affect your labour. The same situation you described I've seen happen at our hospital (I'm an OB nurse). Even when everything falls in to place during an emergency, and when we move as fast as humanly possible, terrible things happen. I'm expecting my first child, and it's incredibly difficult not to let my mind wander to those situations. But I remind myself on a near daily basis of all the women I've looked after that did fine. The statistics for neonatal death are low for low risk women, but I understand if it's your baby that happens to, the statistics don't matter.
People make the best choices with the information they have at the time. The only thing you can do is get informed and make the choice you are most comfortable with. Ask the birth center and the hospital what their policies and statistics are, and I'd be wary of either if they won't tell you.