That's not how it works. Since these are major international hub airports we're talking about, they're almost always direct flights. (Now, from Florida they wouldn't be, but I'm guessing most of your int'l flights aren't).
This is one of those topics that is really daunting to people at first, but once you spend 30 minutes playing around with ITA's multi-stop feature, it's pretty easy to figure it all out. From there, you can manipulate other variables to drop the cost and get exactly what you want. Once you learn the ins and outs of stopovers and open-jaw flights, a world of low(er) cost airfare is opened up.
In fact, with 5 minutes of work (the second search I did), I just found a $634 itinerary to Shanghai and Tokyo out of LAX for next month...
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Oh, and totally agree about flying foreign carriers. I always do codeshare for the miles, but will try to find a foreign carrier whenever possible (I do try to avoid Air China, but wouldn't rule them out--I'd just rather fly Singapore, JAL, ANA, etc.). Of domestic carriers, I usually fly Delta.
Despite their reputation among snobby business travelers, SWA is probably the best domestic airline out there today now that the legacy carriers are engaged in a race to the bottom.