I took a look at the specs for the Korg models mentioned above, and my take is that, when compared to the Yamaha PSR-EW410 or PSR-E463, the Korg might have an edge in some of the technology, such as more polyphony, more memory for onboard recording, and more digital effects, but the Yamahas have a strong edge in flexibility. Unless the Korg specs I read were incomplete, I saw no way to modify the sound other than the digital effects, whereas the Yamahas (the EW410 and E463) have audio filters and an envelope generator for synthesizer-like control over the sound, as well as digital effects like flanger and phaser. And while the Korg can split the keyboard (which, of course, the Yamahas can also do), I did not see it mentioned in the specs where it can layer, or combine two sounds at once for the same notes being played. That is a very nice feature on the Yamahas. Also, the Yamahas have live-control knobs that allow easy, real-time changing of these parameters -- even while playing a song.
And of course, it comes down to what you really want to use the keyboard for. While the EW410 does have 76 keys, they are not weighted, so neither of these Yamahas simulate the feel of a piano. But they are good for a synth/organ feel -- not as good as they should be, but better than recent Casios, in my opinion. I have no direct experience with the above-mentioned Korg models.
As for going with the more expensive SX models versus the E463 or EW410, again, that depends on what you want to do and what your budget is. The SX700 is about 3 to 4 times as expensive as the EW410, and about 5 times as expensive as the E463, but of course, you'll get more voices, more styles, and more flexibility overall.
I have one of the PSR-E463's predecessors -- the PSR-E433 -- and I am very happy with it. I've had it for over six years, and while these keyboards are marketed as "entry level" keyboards, I have been playing keyboard for 42 years.
One more quick note -- I agree with the concerns about buying the PSR-S670 as it is a five year old model that may get replaced soon. Well, the same is true for the EW410 and E463 -- they are two year old models and would usually be due for replacement right about now, but Yamaha seems to have altered their product scheduling. However, they have now put the PSR-I500 on their US website, which is an E463 derivative focused on Indian music, but it has other features not available on the EW410 or E463 that can be used in a wide variety of music. That model may also be worth a look. I know I'm curious about it!