Whether the railways did keep going in the old days in these conditions, using snow ploughs or just men with shovels, I'm not sure. It's the kind of question that needs proper research to answer it, since popular conceptions of what happened (and even memories of those old enough to remember) an be very misleading. But I think it's fair to be disappointed that modern trains and track systems have not been better designed to cope.
Snow happens: it's a foreseeable component of the railways' environment. So why are trains etc. not designed to cope? Your and my favourite train requirement (OK, the only published one), the
IEP▸ one, says:
TS1845 The IEP Trains must maintain Full Functionality during and after running through snow up to a depth of (above rail level):
• 200mm; continuous operation with no speed restriction; and
• 300mm; continuous operation is required but reduced speeds are permitted
Given that the rail itself is almost 20 cm high on its clip, that "full speed" depth is well over a foot - far more than snowfall we had yesterday. Presumably that requirement reflected industry views of current normal practice, so what went wrong?
I can see one thing straight off - snow isn't flat, and you don't want to go steaming (or dieseling) into a drift at 125 mph. Nothing in that requirement says how the driver will get to know what snow depth ahead is, nor even whether that should be the absolute deepest value. So it's not a very useful requirement, really.
ATOC» 's "Key Technical Requirements for Rolling Stock" (2013) does mention snow, but only to say it should not be allowed into various places.
But the point stands - should trains and railways not be more capable?