Part of the problem in this thread is the careless use of the word "connection". OP Is not "connecting" because he is on separate tickets.
The consequence of that fact is that he, not any carrier, bears the risk of a no show at JFK. It also means that if OP has travel insurance, he will find that very few policies cover such a no show, once again shifting the financial risk to the passenger.
What none of us know is what carrier OP is booking xJFK, whether his tickets are flexible, whether there are change fees, what those fees are and what the consequences are of a no show. Is this a quick vacation to London where there are multiple options or is this an important meeting somewhere with only one open flight for the next three days and a change fee + fare difference of $700?
So, all about risk tolerance.