That time frame is when the first bunch of PCT hikers go through, so it's quite doable if you are comfortable traveling on the snow and have some good route-finding skills (trail may be snow-covered and harder to follow).
You'd reach MTR about the 9th, so the odds are they will be open. You'll have a pretty good idea of the snowpack status far enough ahead to make alternative plans for the resupply if need be - as long as you are prepared to spend some extra time or money or both to make it happen.
Stream crossings can indeed be dangerous at that time of year, but there aren't a lot of bad ones on the JMT.
I looked over your list - here are my comments:
I'd find some lighter camp shoes if you feel you need them. Try Sprint aquatics - mine are 2 oz for the pair.
I'd skip the down booties. There may be plenty of snow on the ground, but the weather will be pretty warm.
Driclime windshirt is heavy for what it provides.
You shell is rather heavy - lots of lighter shells and some for pretty cheap, like a Marmot Precip, 12 oz and often found on sale for $70.
Gloves - You sould be fine with just some light gloves that time of year - I like powerstretch glove liners at 1.5 oz for summer in the Sierra.
Lighter gaiters will do just fine - look at Dirty Girls.
Nalgene bottles are heavy.
If you take crampons, get the lightest ones you can, because you will carry them a lot more than you'll use them. Camp has some very light aluminum models. Same for the axe. Camp has one that's about 8 oz.
Are you planning some climbs along the way? You won't need the other climbing stuff just for the trail, even if there is lots of snow.
I'd get something lighter to replace that leatherman.
Otherwise it looks good.
In general, you should be aware that you'll walk on a fair amount of snow, and that there will be lots of water around and on the trail, so your feet will be wet. Be prepared for that, and test out your footwear accordingly to be sure your shoes and socks work well and are comfortable when wet and will dry quickly.
You should have no problem finding dry spots to camp and rest on.
Weather is usually warm in June, but as always it may vary a lot.
Bugs may be horrendous, very bad, or just bad down in the canyons. I'd bring a headnet for sure, and plenty of bug juice. You'll likely be able to escape bugs by going higher where it's snowier, unless it turns out to be a low snow year.
Have a great trip!