Yes, some all seasons are poor in the snow but not all by any means, just not as good as a dedicated snow tire. They just aren't made to excel in deep snow and icy conditions. The next step up is an all-weather tire which gives an increase in snow traction and carries the TPMS (Three Peak Mountain Snowflake) designation on the sidewall. Next is a dedicated winter tire which would typically not give the traction, handling and treadwear of the all season tire but handles deep snow and ice a lot better.
Overall, no one tire does it all to a high level. It's a balancing act blending the attributes to give acceptable performance for the driving conditions the tire is intended for. I've noticed that generally in the best snow traction tires the wet,dry traction, handling , ride comfort, noise and treadwear suffers to some degree compared to the better all season tires.
All-terrains are a viable option.
Through the years i have went through winters with snow tires(Nokian and Michelin) and also with All-terrains.
For winter my preference is
Nokian snow tires
Michelin snow tires
Firestone Destination AT(now AT2)
For winter where I didn't have time to switch the winters on, I have run the Firestone all terrains with fine results in snow/ice. And the new Destination AT2 design have been relatively quiet on highway drives at 50-65 mph. As my Nokians are about burnt, I'm probably gonna run the all-terrains this winter.
Pirelli also had an all-terrains which has high ratings and customer feedback. Noise ratings were mixed though.
After research my picks(from previous thread) for all terrains(that had decent all around performance with low noise based on ratings and customer feedback) were:
Firestone destination AT2,
Falken Wildpeak AT trails.
Yoko Geolander GO15
Goodyear Wrangler Trails.
Pirelli scorpion AT plus(noise?)
Check the above all terrains out as well as all season Michelin cross-climate 2.
* My parameter was a low noise AT tire that also had decent performance in dry, wet, winter and off-road conditions. The Pirelli had a more aggressive tread than the others hence why noise reviews may have been mixed.
Ime and imo, both snow and some AT tires feel alot safer and react better on snow/ice than any all season.