A budget winter driven sensibly is going to be leagues ahead of a summer tyre in the icey and cold conditions in terms of grip and safety.
Because winter tyres have so much movement and are so soft they dont tend to be great for actual handling or fun driving, but that's sort of the point, if the conditions justify winter tyres you're meant to be taking it easy anyway.
For the best results a good brand like Yokohama or Michelin etc is going to be the way to get the best performance from your winter tyres, but as a stop gap to get you safely through the winter while driving carefully a cheaper winter tyre will be a safer option that a summer tyre.
The year before last I had summer tyres on the back of a 200SX, and in the snow the camber of the road was enough to drop the back end into the kerb, repeatedly. I had previously never understood all the fuss, as I had never had proper sporty tyres in snowy conditions that had so little give due to tiny sidewalls, and they simply didn't work. Switching to proper tyres meant I could actually get around, and round here with all the hills the difference was huge!