This is not a logical necessity, but it is likely. That uneasiness has created a tension throughout the history of America. In terms of moral philosophy that was indeed a revolutionary declaration. It is sad that the American people have not yet fully subscribed to it. The pursuit of rational self-interest of course creates benefits for others, as earlier classical liberals defensively argued.
Human beings enjoy a basic harmony of interest.
Specialization and the division of labor bring a level of general prosperity otherwise unattainable. But morally speaking, that is strictly a secondary issue. We must persuade our fellows that capitalism is good because each person is an end in himself who is justified in pursuing his happiness. Until that job is done, we will fight an uphill battle. There is no third way, as there is no middle course between life and death.