Distance only became a factor because of the length of the war, which only occured because of the strategic vision of George Washington. Also, distance did not play as vital a role in the British defeat as did the will of that country's government to continue the fight.
Popular support? There was no overwhelming majority in favor of the Revolution amongst the Americans. A plurality yes, but not overwhelmingly larger than the Tories or those who simply didn't care one way or the other.
Also, don't overestimate the support of the French. The only battle they played a truly decisive role in was Yorktown, but Cornwallis wouldn't have retreated to an earlier-than-usual winter encampment if not for the weariness inflicted by the prolonged war.
Hannibal, Napoleon, etc. fought with regular armies that were equipped and trained by regular means. The American army had to make do by any means possible, and much of that came through the leadership of Washington.
As far as being a "mediocre corps level commander," he wasn't. A mediocre commander could not have survived and persevered for six years with an army that was constantly outnumbered, outgunned, and encumbered by the Continental Congress' constant bickering amongst themselves. Washington was betrayed by his ablest general (Benedict Arnold) and played the role of an Eisenhower in balancing the personalities of numerous generals both foreign and domestic. He devised tactics to give his army a chance to survive battle and even win some, often defying the standards of the day for what constituted battle. He conceived of movements to minimize the long odds against his troops (think of the crossing of the Delaware).
Washington was a brilliant man, a great student of any subject he put his mind to, and through his sterling character became an inspiration to all Americans. Think of it this way: If Grant had failed, the Civil War would have lasted a few years longer but the Confederate army was already mortally wounded at Gettysburg; if D-Day had failed, we would have licked our wounds and tried again in a few years, but we already had a toe-hold in Italy and the Russians had the tide turned in the East. If Washington had failed, there would be no United States as we know them, and we'd still be speaking...English, I guess. ;>)