Kennedy may have grown up with privilege, but he was never accepted in the higher reaches of Boston society - the world of old Back Bay and Beacon Hill, the old WASP families like the Cabots, the Lodges, the Peabodys, etc.
General anti-Catholic prejudice was almost universal in the US in the 19th century and persisted, though diminished, well into the mid-20th century. 19th Century anti-Catholic feeling had some basis in fact, in that the Roman Catholic Church was then incredibly hostile to liberalism (in the 19th century sense of promoting economic and political liberty and religious tolerance) and was a deeply reactionary force opposed to the American experiment. Most Americans in the 17th-19th and 20th centuries until near the end, were either Protestants (if churched) or of Protestant stock though unchurched.
In the late 19th and first half and a bit of the 20th centuries, in many cities (where most Catholics lived) close to a majority of Catholic children attend parochial schools and then Catholic colleges. It was possible well into the 1960s for a Protestant or Catholic to go through most of their lives and have almost no social or school contact - other than impersonal contacts in business or on the street - with the other faith.
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Joseph Kennedy earned his money running alcohol during prohibition. Is it any wonder this family was not accepted?
Would you be welcoming to a drug dealing family moving into your neighborhood?
Seems to me that the rejection or acceptance of the Kennedy family had little to do with religion.