Bigsbies don't have tuning issues. That problem usually lies with the nut. I've had 4 guitars with Bigsby vibratos. From the cheapest B700 imitation to a US V-cut B3. All tuning issues were caused by the nut, never the Bigsby.
The nut is indeed quite often the culprit wrt/ tuning issues - even with fixed bridges FWIW.
If it's range you want, get a Floyd.
There are other issues with Floyds, as well as with any "floating bridge" trems - like having the whole damn thing going out of tune when doing bends or open tuning or breaking a string.
The opposite spring idea has been around for more than a decade.
Yeps - but there's more than just an opposite spring involved in the trem-king.
Ok, I didn't have an chance to test a trem-king so far (but I'd be very interested) and to be true I just don't believe it could be the Miracle Solution To All Trem Problems

. But from what I've seen of the concept, it might really be an interesting alternative to at least the vintage-strat trem.
Just my 2 cents, really...