It's a deceptively tricky DSP problem. The surface issue is that nonlinearities create harmonics that wrap around the Nyquist frequency, which creates aliasing. To deal with it you oversample, which introduces some frequency/phase distortion, and in the cutting edge solutions they run at 8-16x oversampling (which means 8-16x more CPU cycles for the distortion algorithm). And it gets worse with heavier distortion.

On the algorithm side there has been some crazy good work with creating black box models using Volterra Series and dynamic convolution, but it's an O(n^2) algorithm (although I believe there's an O(logn) solution out there). The top of the line modeling amps (mostly Kemper, there are a few others) use these approaches among others to give you hundreds of amps at your fingertips, and they sound great. Plenty of high profile bands use them, Metallica probably the most prolific.

I bet someone has applied convolutional neural nets to the problem, since I'd suspect they're exceptionally well suited for black box modeling of nonlinear distortion. Sub in an atan(kx)/atan(k) for your activation function and each neuron would be it's own first order model of a tube stage. The asymmetry might be more difficult to handle, which is where a lot of the warm tube goodness comes from.

The UI problem is well known. That's really what sets one amp design apart from another. No tube amp is particularly unique in signal path or component choice, the ability to dial in a tone is the secret sauce behind the classic designs imo, also why some newer models from Marshall are so much worse than newer guys on the block, but that's just my opinion.