In terms of why Washington was a "good choice for commanding the [Continental] Army, the simple fact is that no American had more military experience than Washington. He had developed a reputation (both good and bad) as a military man during the Seven Years' War. Afterward, he had applied for an officership in the British Army but was rejected. Of course, it helped that he silently campaigned for the job of commander of the army by wearing his military uniform to the Continental Congress.

Secondly, Washington was a good choice for commander because he was a Virginian. The most radical Continental Congress members by June of 1775 (when Washington was named commander) were from Massachusetts. After all, this is where the fighting was taking place. And the idea among radicals was that choosing someone not from New England would have a unifying effect and that it would dispel the perception (held by many) that Massachusetts was dragging the other 11 colonies (Georgia was not represented at the Congress) into war with Britain.

Washington's primary contribution to independence was not in establishing it, since he played no significant part in the Continental Congress debates. Rather, his contribution was in securing independence, which he did primarily by winning the war. In that sense, his greatest military contribution (and, hence, his greatest contribution to independence) was devising the Continental Army's long-term strategy. Washington and the Army had suffered debilitating defeats in New York in the fall of 1776. And it quickly became apparent to Washington that he could not beat the British in a traditional, European-style war in which battles were fought head-on by columns of soldiers. After subsequent defeats at Germantown and Brandywine the following year, Washington realized that he didn't need to defeat the British in battle to win the war. He merely needed to avoid total defeat.

The British employed a European-style strategy that prioritized capturing major cities. This, however, could not itself end the war in the colonies because they were so large. Washington realized that the only way the Americans could lose the war was if the Army was captured (or disintegrated). The British could capture New York, Philadelphia, Charles Town, and other major cities but as long as the Continental Army was still out there, the war was not over.

Washington's other insight was that, now with the French declaring war on Britain, they could no longer afford to prioritize the American conflict, and that fighting two wars in two different theatres would take a toll on Britian's population and its fiscal health. Therefore, he just needed to keep the war going until the British people tired of supporting it, both in terms of men and borrowed money.

So, Washington's most valuable contribution to independence was securing it by winning the war, which he did by developing a long-term strategy designed not to win but to avoid defeat.