Instead of using capacitors ... why not use a negative supply rail? Then you can switch the direction of the current in a coil using only two transistors.
The capacitor circuit has the advantage that could can't mis-drive. If a transistor of the half bridge stays turned on for longer than the msec required, nothing will happen. The half bridge chip (usually) already prevents that both transistors are enabled at the same time. This is great when you develop firmware and might end up with one pin staying high for too long.
With a direct drive, the flip dort magnet wire could quickly burn through if you leave half an amp for too long.
It is still going to be many amps.
The capacitor circuit has the advantage that could can't mis-drive. If a transistor of the half bridge stays turned on for longer than the msec required, nothing will happen. The half bridge chip (usually) already prevents that both transistors are enabled at the same time. This is great when you develop firmware and might end up with one pin staying high for too long.
With a direct drive, the flip dort magnet wire could quickly burn through if you leave half an amp for too long.