Horrible tuning problems are the reason the Bigsby went out of fasion IME.
As I pointed out, it's old tech. My Bigsbys stay in tune, but then I use relatively heavy strings (.011-.052, wound 3rd) which in fact are LIGHT compared to what the Bigsby was designed around. Some of those old archtops had a wound 2nd! For somebody brought up on .009s and one spring on the whammy, they are difficult. But as long as you don't mind the llimitations, they are fine. Just don't expect a Bigsby to be "perfect intonation forever" because you have to work with it.
After all, it's like my Dobro: 1930 design, no pickup, how basic can you get? But if you are accustomed to making your own sound*, as opposed to having the machinery make your sound for you, the Bigsby and the Dobro have their place.
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*The corollary to that is that if the machine makes the sound for you, it's not YOUR sound.