A little more detail- super steep terrain.
On soft snow, when descending I can weight my heels as much or as little as I like and therefore either "stomp, stomp, stomp" down the hill or glide down.
On hard icy snow, the shoe doesn't sink in. You either rely on the snowshoe's crampon or take the snowshoes off and use conventional crampons. There is way more traction with crampons than you get walking on dry ground. You can walk straight down a very steep slope. With your foot parallel with the ground, that means there's a lot of force pushing your feet into the front of your shoes, and pushing the shoes to the front of the snowshoe or crampon binding.
So on hard, steep snow I have been much less happy using trail runners inside NEOS overshoes. My toes got jammed to the front and squeezed by the snowshoe binding. Switching to crampons the straps squeezed down on my foot and were uncomfortable. After trying that once coming down So-California's Mt. Baldy (4,000' of descent in about 3 miles), I don't wear runners when I might need crampons.
Using either mountaineering boots and gaiters, or hiking boots (with a pretty stiff leather toe box) inside the NEOS I didn't have this problem.
For the OP's question about hiking around the California Sierra Minarets, consider the terrain. If you keep to the moderate terrain of Mammoth Pass, Red's Meadow, up the JMT or River Trail route a soft boot or shoe will probably be fine. If you are going into more mountaineering type terrain then a stiffer boot might be in order.