You could try, with the usual cautions, to spin it in the correct direction when starting. Put it on its side, have one person turn it on, and someone else flick it round with a long bit of wood, like a broom handle.
If it then runs, it is almost certainly the capacitor. I've had that happen on one.
Mowers normally don't have a centrifugal switch. If the capacitor is small, it won't hurt to leave it connected, so no switch is needed. The small capacitor gives poor start torque, but on a mower that's fine as the user will just keep the blades away from grass until they're up to speed.