vtsnowedin,
"...perhaps during ice age winters it was dry."
You touch on an important point for the large scale: glacial climate in middle and higher latitudes was cold and dry. During the times of particularly severe climate that saw Neanderthals move south it was even colder and drier. The tundra and boreal forest that occupy those latitudes today didn't exist; grasslands were a major component of the regional ecosystems, as indicated by the predominance of grass and wormwood/sagebrush in the pollen floras from that time.
That's why grazers were so abundant then in areas where they couldn't survive today; mammoth, horse, bison, saiga, wooly rhino (Eurasia only)--the quantity of skeletal remains throughout the boreal realm is staggeringly large.
Horses and bison have been re-introduced to Alaska but they live in areas that are well-drained, not on tundra. Tundras are mainly waterlogged terrain because of permafrost near the surface which prevents water soaking in, and low rates of evaporation due to low temperatures. Much tundra vegetation is well protected against herbivores by toxic compounds (although several kinds of berry are abundant.) There are large herbivores in tundra country, moose and caribou, but moose came in late in the last ice age and are found in lake and stream-side habitat (and, nowadays, along railroads where a bull moose will occasionally stare that train down and learn too late), while caribou are lichen eaters with large feet that allow them to traverse tundra.
After one field season on the stuff I swore I would never work on tussock tundra again.
Moose, by the way, are expanding northward in Alaska, presumably due to regional warming.