Keep in mind that it is impossible to give a satisfactory answer to your question. A lot will depend on conditions when you actually reach the snowfields in question.
There is also a heck of a difference between 9000 feet in California and 9000 feet in, say, British Columbia. There is also quite a lot of room in interpretation of the term "large". The other question I'd have is how steep are these snowfields?
Having said that, I've done quite a bit of on-snow travel in the mountains of Oregon, Washington, California, and British Columbia on running shoes. Sometimes if I am expecting well-frozen snow I will take Katoolas or something similar. Generally I just go with the shoes and hope for the best. In a few cases that's forced me into some awkward detours, but 99 percent of the time it has worked out fine for me.
Traversing a snow slope is an important subset of "sidehilling". Regardless of footwear, sidehilling is more tiring than walking on a flat surface and produces more wear and tear on ankles, knees, and hips.