Indeed you can get "avalanche diodes" for the same purpose i.e. rectification with a bit of surge immunity. Though I don't think these are common for almost anything at all beyond alternators, where they clamp load dump and so are available in much lower voltage ratings. (Even then, I don't think you see them often from general suppliers, they're very much an OEM thing?)
Oh hm, I wonder if TVS have lower Vf, since they can have thinner junctions -- you can't get a 1N5400 as such anymore, they're all massively overrated compared to what they're sold as, that's why 50-400V diodes have the same Vf and Cjo, then 600-800V are actually 1.2kV or whatever, etc. Still won't be as low as a schottky of the same value, but might be a few mV of justification in that rare situation where you have some to spare.
At 400V, I doubt Vf is much less though, and yeah, you'll be using them like pretty much any other diode, hardly a difference in Vf, [forward] surge, etc. Speed is probably faster than "general purpose" but not by a whole lot for those voltages, I would think. Not sure about softness. Capacitance will be higher, of course. And that's basically it; for general purpose applications, yeah, no problem.
Another good application for zeners as rectifiers, is in current-sourcing supplies, like an offline capacitor dropper supply. Or a current transformer maybe, when you need a DC feedback signal and you don't have dual or quad diodes handy for rectification.
Tim