I've been to the Winds twice in late June. The first time was perfect - no bugs and only the passes were snow covered. The 2nd time (last year) was still a cool trip, fun and without bugs, but snow was absolutely everywhere. It was a bad mix of tons of snow and very hot temps. There was no firming up overnight. Even with snowshoes on I'd sink to my waist at the crack of dawn. Fresh avalanche debris littered many slopes, and I witnessed a few cornices collapsing (one directly above a route one of the workers from the local sports shop had taken hours previously). I had planned a week-long backpack from the Cirque of the Towers up and around Mt. Hooker, but instead ended up making a few short trips, to the Cirque, to Eklund Lake (not sure of the name), and to climb SquareTop. Even Squaretop, typically a class 2, was a serious adventure given the too-mushy-to-self-arrest conditions.
So, I'm guess what I'm saying, is it all depends on conditions. I'll be heading out there again this summer. My plan is to call the local sports shop a few days before I leave and get an update on conditions. If they're good, I could be in the Winds as early as June 20th (would love to team up with someone to climb Gannett, by the way). Otherwise, I'll head elsewhere and try to hit the Winds in August.