Oh, it's doable; living in Detroit, the winters of '73-'74 and '74-'75 were spent driving a '73 MGB Tourer__still on showroom tires! I too did the 190* thermostat, but I had the entire oil cooler wrapped in cardboard, and all but the top 3" of the radiator covered too.
If there was one (1) thing I'd say to be careful of, it's watch your right foot! Especially true when you're forced to solo pushing the car by the steering wheel and A-pillar; I bet I ran over my foot at least three (>3) maybe four (!) times! Fortunately, I was still immortal back then, made of rubber, and I never let the car get away from me (albeit, there were a couple of close ones).
MGBs are probably only marginally less rust-prone than a Healey, so all the same caveat's apply__not that I was worried about the car's longevity forty-four (>44) years ago (I wonder where it is today__if it survived...).
This picture was from 1996__different MGB(GT) altogether, and just visiting "home" from Louisiana__at Halloween-time, so minimal exposure to those unsavory elements!
Now some years later, I did get serious about driving a sportscar all winter in NW Ohio, but it was outfitted with dedicated snowtires, traction control (which you actually had to disengage in deep snow to make any progress) plus a hardtop, heated seats and a real honest to goodness heater/defroster! This lil' babe has been through twelve (12) years worth of snow, ice, salt & brine, and is still living__now a leisurely life in SW Florida__to belie any evidence of such abuse! Oh sure, the bare aluminum castings are a little fuzzy, and the cad-plated hdwr is now brown, but the galvanized sheetmetal and plastic-coated undercarriage is absolutely corrosion free (I've had the bolt-on rockers off and was SHOCKED, expecting to find at least a little bit of rust, but there was none).