It snows far less on White Mountain, so I wouldn't be worried about snow. It's a 14 mile round trip from the locked gate, but not a lot in elevation. A big one day hike for kids without experience. You need to be acclimated, because this hike begins at 11k feet and it only goes higher from there. But it's easy compared to Whitney. You could ride mountain bike to the top, given the "trail" is a jeep road. No water anywhere up there unless you can melt some snow, so you have to pack quite a heavy load just to stay hydrated.
Whitney is more challenging in summer, but you have the added thrill of climbing the highest mountain in the lower 48, and it feels much more like a big mountain compared to the more high altitude tundra appearance of the White Mountain trail.
Permits for a larger group are much easier to get on White Mountain. Perhaps do some lesser peaks in the Sierra as warmup hikes to acclimate? First a few laps around Tuolumne Meadows, up to Clouds Rest maybe, then a visit to the top of Mammoth Mountain (on foot), or do a brief multi-day look across Koip pass and Koip Peak, and back via Donohue Pass and Lyell Canyon. I never did White Mountain without spending at least a week in the Sierra, so I don't know how tough it can get when you're not acclimated.
One thing to keep in mind about White Mountain is that if one in the group is getting in trouble with altitude sickness, there's no easy way down to lower elevation other than to get all the way back to the road gate and drive for a while to get lower. If you descend on foot cross country down the side of the mountain you are going into rough terrain and no-man's land without water.
