Why did more Native Americans side with the British than with the Patriots?
The British promised the Native Americans that they would control the expansion of settlers into Indian land.
The British made many promises to the Native Americans to convince them that they should provide support to the British government and military in the Revolutionary War. Mostly over-promising the British provide mere trinkets and low money offers to the Indians who were not aware of they way they were being mislead. In saying this; it does not mean the Patriots were treating the Indians any better.
Trade and Land. The British promised trade and protection of the Indian's land. The American Colonists kept pressing westward and north into the Indian lands.
After the French and Indian Wars The British in 1763 drew a line along the Appalacan Mountains making it illegal for the Colonists to settle west of the mountains. This made the Indians happy and the Colonists furious. The Colonists did not honor the law making the Indians angry with the American Colonists.
After the French and Indian Wars the British took over the fur trade from the French. The fur trade was important to the Indians bringing in desired manufactured goods. ( and sadly alcohol ) Many Indian tribes transferred their loyalty from the French to the British.
Other Indian tribes such as the Irqouri had fought with the British against the French and the French Indian allies ( traditional enemies of the Iraquois ). These tribes continued in their allegiance to the British Crown when war broke out between the British and the Colonist.