http://www.npri.org/docLib/20121003_JFK_speech_at_Notre_Dame_searchable.pdf
JFK warned of federal government absorbing too much power at hands of states.
He also believed in tax cuts as an economic stimulus to counter recessionary pressure, and he was much stronger on national defense than he is given credit for.
But you're missing a much larger point, and it goes to the problem of historical knowledge.
JFK came from an era of centrism where the Left had a coalition of southern evangelicals, and the Right had liberal Rockefeller Republicans.
DURING this era of centrism Eisenhower supported FDR's New Deal and was a staunch supporter of Big Government infrastructure projects. He believed in
government jobs as a way of building this great nation and putting hard working families to work. Do you know Eisenhower's top tax rate? Or Nixon's, who created the EPA and was also a believer in government's capacity to build infrastructure and put struggling Americans to work. These men believed that great things could be accomplished by taxing the wealthy, and they proved it.
You are advised to study the policies of Truman through the evangelical Carter. You will see a period of centrism where nearly every president supported policies from the other side.
We didn't see radical partisanship until the Reagan Revolution, which didn't fully blossom until 1994 with Newt's takeover of the House. This ushered in a new era where the Republican Party was purged of moderates. Centrism died, and any Republican who was caught not obstructing a Democratic president was removed by the Primary Process.
George HW Bush was last great moderate. I voted for him, and I don't regret it. I came to disagree with many of his policies, but he raised taxes when he had to, and he conducted foreign policy with dignity.
Read My Lips: If he ran today, he would not be allowed near the Republican ticket.