3mm is the legal minimum tread depth here, but the recommended minimum tread depth is 5mm (.2 inches). At 5mm they should work OK on ice and clean pavement, but even at 5mm you’ll experience decreased performance in snow.
I’ve been driving on studless winter tires made for Nordic climate for ~10 years now (and on studded tires for 20 years) and I routinely experience markedly decreased performance during the third winter (four different cars, three different brands of tires during those 10 years). The tread is a lot deeper than 5mm after those three winters.
ETA: The problem isn’t so much the tread depth as the dulling of the studs or rounding of the edges around the sipes. Those two things kill the tire’s grip