I just took my Behringer overdrive pedal apart, which is basically as cheap as they come (I use it for vocals occasionally, I'm not a guitarist, so I figured why make a huge investment).
Inside is, as I suspected, an op-amp and some diodes, albeit a few more support components than I'd expected.
A digital distortion circuit would require far more complex design and more expensive parts just to replicate what can be done for peanuts with a handful of analogue components or an analogue ASIC.
I might be letting theory trump practice again, but the only time I can see it being sensible is in a multi-fx unit (and in my experience the distortion mode always sounds awful), or when you need total flexibility over how the distortion is happening, e.g. virtual models of 25 different circuits.
That said, the Behringer multi-fx pedal I purchased at the same time, that
is digital. I use it for chorusing the vocals and I have no qualms about using that in an otherwise all-analogue setup. Besides, it achieved precisely the effect I was aiming for, the moment I plugged it in

Unless you can get a really nice spring reverb or can afford an actual plate or chamber, the chances are you're going to be using a digital reverb in the mix anyhow.
But digital
distortion, ewww