After the Opium War, Britain and France signed treaties with China that said they could establish "spheres of influence" over parts of China. Britain and France went this way, but why didn't America?
American attitudes toward China were best expressed in the Open Door Policy. This policy was, in fact, aimed at "rolling back" some of the special privileges others were trying to "rent."
American didn't want to "rent" parts of China because she didn't want other countries to "rent" (and thereby divide) China into 5-10 "special" regions.
America was on its way to becoming the world's foremost commercial force, and as such, preferred a unified, "open" China with a (theoretically) level playing field that would benefit her more than other countries.
Put another way, America preferred to have equal access to all of China, rather than special access to part of it. (But "some access is more equal than others.")
This French and British fleets were government-sponsored fleets containing official Navy ships. Their expeditions to China were conducted as part of official government policy with expansionist motives.
The large majority of American ships involved in the China trade were private merchantmen, not Navy warships. The Americans had a few isolated warships in the area, such as the USS Portsmouth, however those ships were only there to protect American vessels, not embark on an expansionist policy. They had orders to maintain American neutrality and the Americans moreover refused to make an alliance with the British and French in the wars, because we had a policy of neutrality and non-interventionism.