Same here!With four dogs it can be quite a chore. Many years ago, the problem was solved by putting a small,fenced in area directly out of what became the mud room. With three well placed spotlights on the second story, it's like daytime.
The only snow removal I do is near the entrance. When the dogs air, they are not left alone and I "pickup" right after every airing......no matter how late, cold or inconvenient the weather may be.
In the warmer months when paws are dirty, they walk around in the mud room and I follow with a mop full of bleach water until no more muddy prints are left. They then cross over a thick rug (which absorbs most of the water on their paws) and re-enter the house. It's a "dirty job" simplified (and workable).
When spring comes the yard has no "surprises", the grass grows back quickly and mud is not much of an issue......but still the yard is ALWAYS picked up immediately.
It has become so much of an easy habit, I hardly think about it.
edit: We have had as much as 15-20 inches of snow (but that is rare). Lake effect snow (deeper than that) would probably mean using a snow blower.