I believe that the tally was a 8,800 vote margin in IL, but your point could still be valid. However, based upon what I have heard, I think you folks should look to the WV Primary, when Kennedy beat Humphrey by about a 60-40 margin. Back in 1960 not too many people voted in primaries and thus the union vote was critical. Word has it that it was Kennedy, Sr. who managed to get the unions to endorse his son. As to the general election, TX reportedly had vote fraud as well, especially in some of the counties along the Mexican border, where reportedly one county reported more total votes than the number of people who were registered. However, the TX margin was about 50,000 and it seems much less likely that vote fraud made the difference in TX, like it well could have in IL. Besides IL having a closer vote difference, it also had more total votes than TX, thus meaning that the percentage difference was even closer. KY reportedly had vote fraud in favor of Nixon, however, Nixon won KY by enough of a margin that such probably probably mattered little there.
Nixon probably erred by choosing Lodge. If he had picked Sen. Dirsken of IL, he would certainly have carried IL. However, he still would have come up short. There was no one in TX that he could have picked, to carry that state. If somehow he could have carried IL and TX, he would have won the election by just one Electoral Vote. He lost Hawaii by a few hundred and NV and NM by a few thousand each. More important, he lost MN and MO by 10,000 each, as well as SC for that matter. The 1960 Election will probably never be repeated in terms of the large number of states that were quite close or relatively close. I think NJ was only about a 20,000 vote difference. I suppose that if we forget TX, Nixon could have probably won by that one Electoral Vote if he could have carried NJ and either MN or MO, assuming IL as well. Many claim that the first debate made the difference, though those who heard it on the radio gave it to Nixon, while those who watched on tv thought Kennedy won. Many experts say that Nixon lost that first debate, mostly because he refused to wear makeup and that as a result, his appearance was not good. As close as the 1960 Election was, it is quite possible might have won if he had put on makeup in the first debate, though we cannot be certain. It does seem likely that he would have carried IL and maybe Hawaii, but uncertain about all the other close states. What does seem certain is that the 110,000-120,000 vote margin in the Popular Vote for Kennedy would probably have shrunk to a virtual tie or a slight Nixon lead.