Northbeach:"If you had a fully charged 50 amp hour battery (meaning it will deliver 50 amps for one hour or one amp for 50 hours)"
That is not precisely correct. A 50 amp hour deep cycle battery is rated at the amp hours it can deliver over a 20 hour even discharge cycle to 10.5 volts. In this case it can deliver 2.5amps continuously for 20 hours before dropping to 10.5V (fully discharged).
It will NOT deliver 50 amps for one hour. It will deliver one amp for more than 50 hours.
Since the quoted 3.3amp TILLER Pilot draw is HIGHER than the rated 2.5 amp 20 hour draw...it will NOT last even the 15 hours you quote. This is often an overlooked area when considering sizing battery or alternator capacity needs.
The rule is: If your hourly amp draw exceeds the battery capacity divided by 20...you wll get less than your rated amp hours from your battery and should size up accordingly or plan on recharging more frequently.
That said...I totally agree with the rest of your post.