no you don't need the hand throttle ... it sounds like ya got it right, keeping in mind there's adjustments possible the pedal doesn't move real far [you could have someone put it thru it's travel and watch the carb to see if it has full sweep] .... the fuel problem could be junk in the line, especially if the tank isn't new or renewed, have an inline filter on it? might also have been a poor diphragm in the pump, that decided to split, or could be a worn pump push rod that finally got to not pushing enough to pump anymore
Bill