It could have played out something like this.
Madeleine exits apartment after being freaked out by an intruder attempting to get in through the bedroom window, runs out into street and is knocked down by a passing motorist. In the gloom and with nobody about and seeing that the child is lifeless the driver panicks, lifts the child, puts her in his vehicle and drives off to consider his next move. Maybe he doesn't have a licence, maybe he is underage, intoxicated, has no insurance, a banned driver, out on the rob, the possibilities are endless.
After a delay his conscience gets the better of him, he then decides to take Madeleine for treatment on the off chance that the medical centre is open so returns to the town and drives down to where he can park. He then carries the child down to the medical centre passing the first of the Smiths before realising it is closed. Terrified he walks on so as not to attract undue attention, he passes the rest of the Smith family and goes down the steps to get out of the way only to return the way he came and return to his vehicle. Realising the girl really is deceased he reviews his options and drives off.
And so the Madeleine McCann mystery is born.