That Daley gave the election for JFK is disprovable by looking at an electoral map - he'd still have won if Nixon took Illinois.

There's evidence of voter fraud, but there were always rumors that the counter-allegations of downstate GOP fraud would have turned it into a mutually assured destruction situation. There was a particularly high voter turn-out, but that's what the Machine does did - you got your people to to the polls to vote for your guy. There was a narrow voter margin, but that's Illinois for you - it's a red state with a very blue corner, and sometimes the question is which side comes out. What research has been done on the fraud hasn't found game-changing fraud, even with the narrow margins.

I hesitate to speculate on the Machine's direct influence over national politics during that period, but Chicago was the stage for those events, most visibly the '68 convention, but more subtly and more profoundly the Chicago Freedom Movement. This was what amounted to King's march on Chicago, and the press for housing reform. There were limits to the white north's belief in civil rights, and Chicago is where they got made very clear. To the extent that was MLK vs. Daley, Daley won. There's some local mythology of the very Chicago Way that it played out that's exaggerated (and a little racist), but it did play out there, and (like '68) affected the tone if not making a material difference.