Doing these kinds of searches is the next great business opportunity in online travel. When you buy a flight ticket, what the airline is really selling is a fare - a pre-made package of one or more flights, on one or more partner airlines, with specific restrictions and price. QPX (and all other flight search engines) don't search specific flights - they search fares.
If you miss a connecting flight, you will be re-booked on the next flight, but it all stays within the context of your fare. Not so if you split up your journey into two distinct tickets. However, a lot of people won't mind this.
In your example, you can split up your travel dates, and use part of the $1600 you saved to stay a few days in New York. I like to do exactly this - the cheapest trans-Atlantic flights are offered by charter airlines from London, and I like to make time for a few days stay there when flying from continental Europe.
Not a big chance of missing your next flight with that kind of window. Even if you know you are going to miss the next flight, with most airlines (but very rarely with charters) you can re-book up to an hour or so before boarding, for a fee that may be less than the difference in ticket prices you would have paid anyway.
This also works for round-trip flights. I often book round-trip flights as two different tickets. A lot of the time it is cheaper, but it also offers more flexibility (remember, you have to pay change fee and ticket price differences - and the airline might only have expensive fares available!) in case you want to change one of the departure or return flights (but obviously not both, as then you're paying 2x the change fees).
Of course, what I haven't said yet is how difficult this will be to pull off. Any travel company that does this will be boycotted by all major airlines. Many fares you're bound to buy this way are sold at a loss (better to sell for something than have an 80% empty plane flying and get nothing is one way to put it; there really is no easy way to price flights - most airlines have teams of math and OR PhDs working on pricing). The multi-flight fares usually have a larger margin tacked on.
The only way to solve this problem is really to nuke the existing fare system and start over. As long as "flagship" airlines keep getting bailed out by the government, they will be able to undercut the competition (and themselves) into bankruptcy with the irrational fare arrangements.