The starter solenoid could be faulty, but from experience batteries can apparently take a charge and show 12v but still be unable to deliver the current to turn the starter motor, especially the earlier ones. Can you find an alternative battery to try or even use jump leads from another vehicle? However, fitting a new battery is always a good investment and is much simpler than replacing the starter motor, even if you do eventually have to do that. The earlier starter motors usually fail when hot, not on the first start of the day.
