Enchantments in Early June

Yes, there will still be very deep snow up there that early in the season, and Aasgard will be outright dangerous (the Snow Creek side considerably less so, but the last scramble up to Viviane can be sketch in snow too). If you're not comfortable with snow travel, passing is a wise choice.

You could still go to Colchuck or Stuart Lakes though, since Core permits are good for any zone. They will be considerably easier - Stuart may even be completely melted out.

It's getting harder to predict when the snow will melt out. A few years ago it was an above average year for snow pack and then heat domes burned it off very quickly. Which created hazardous creek crossings. It depends so much how the next couple months play out.

Aasgard can be dangerous in snow. The section above Vivian too, it's rounded granite slabs, they grip well when it's dry but would be terrifying in the snow. Forest Service put rebar into the rock after someone died there.

Overall, it's probably not a good idea unless it's an exceptionally hot spring.

I’ve been up there mid June one year in the snow and there were parts that were super dangerous. Would never go that early again!!

It has not been an above average snow year in the Cascades. We're right at normal. https://www.wcc.nrcs.usda.gov/ftpref/data/water/wcs/gis/maps/wa_swepctnormal_update.pdf

That said, other commenters are correct. There will almost certainly be very significant snow in late May which can make things dangerous. Colchuck and Stuart are both excellent options.

We won the lottery too for the corezone and contemplating of doing the hike to Aasgard. Although we have hiking experience during snow and cold season in Canada but I’m scared about hiking to Aasgard because of the informations I found that it could be dangerous because some part of the falls(?) or river is covered in snow and melting

I wouldn't go up there in June unless the snow melts off suddenly and rapidly like it did a few year ago. If the snow is in transition, it means that the creeks are high and the snow bridges are collapsing. Having fallen through a snow bridge on the way to Nada Lake, I highly do not recommend that (I was fine). Although the core would be quite safe, there's a lot of talus and rocks whether you approach from Snow Lakes or Aasgard side. If you want to go into the core when there's snow, you really want to go when the snow is consolidated and avalanche danger is lower, but before it starts melting. But that's mountaineering. Camping at Colchuck Lake is nice though.

Your analysis is correct. Sorry! Enter again next year and choose dates mid-July to mid-September.

Don’t try it unless you know exactly what you’re doing. But much of the area is beautiful and might be accessible. Check the WTA website a couple of weeks before your date.