I was kind of joking about Shasta as it is a big outing with out lifts. That said, I think the spirit chief is just fine with people skiing/riding Shasta. Hundreds of people do it each year, and there is a guide service. It is considered one of the grand classic summer descents in North America.
Arapahoe Basin in Colorado will be open until sometime in June. Generally the first week is when they close, mostly due to lack of business not snow. I think Snowbird in Utah will have some lifts spinning in June too. Utah got a good pounding of snow, so it's probably a good bet.
I wasn't aware that the Shasta remark would actually be worth considering: I'll look into the guide service you mention.
Thanks for the Shasta, Snowbird and A-Basin references: now it's research time.