Trying to hook a headlight onto my push mower. Apparently the low side of the ignition coil supplies plenty of voltage (~200vac) but very little current to power any light ..without burning up the coil quickly. So I believe I need to add a 12vdc starter off an electric start model. My thinking is that once I pull rope start the mower, the starter will behave like a generator and supply a constant 12vdc power source, whether a battery's connected or not. So my question is; 1) Am I correct that these small push mower starters also act like generators while running by containing the required internal charging coil necessary to recharge the onboard battery ? and 2) Am I correct in assuming I can pull start most any electric start mower without any battery connected at all (magneto/coil supplies spark), or is the battery a critical component of the start circuit on those models ? thx

No, the starter motor will not act as a generator while the engine is running, it does not stay engaged and does not spin while the engine is running.

You do not need an electric starter, but you would need an alternator coil and flywheel designed for the alternator to generate 12 volt current, you can run a light off ac current, so you would not need to rectify the current out of the alternator if you did not want to .