That depends on a bunch of things. Do normal people do it? If they can manage the good sense to bail off trail if it’s beyond their own limitations to deal with whatever barrier that presents itself — sure.
Not everyone is comfortable with cornices, suncupped deep snow, hordes of mosquitoes, deep, swift, very verrrrrrah cold water crossings, postholing, or the possibility of crashing through crusty snow bridges they didn’t even know were there into frigid snowmelt and boulders. Some who don’t even know these risks exist will do it and succeed. Quite a few won’t succeed, and drive up the numbers of rescue efforts in the park(s) and for the Inyo SAR. You might even get away with doing the whole trail unsupported — the usual resupply spots will not open for the season until the snow melts, by the way. A whole lot depends on how much snow is really left in June. Whether the passes open, whether the Mist Trail opens, whether Muir Trail Ranch and Vermillion open….
The real question is, are YOU aware of the risks, and able to mitigate them with YOUR skill set and experience, of which I do not fully know… you do.
I suspect that folks who backpack a lot are not particularly normal…. I know I’m not. If by normal you mean someone who gets a whim to do the John Muir Trail one day, gets a permit, borrows some gear, and just heads up there, I don’t think a normal person can do it in June, no. If by normal you mean someone who backpacks every month, has winter and three season gear (sometimes in duplicate), has training in snow navigation, has been out in the Sierra every year since s/he was six, and knows how to pack a bear can with the maximum amount of food as well as how to properly hang overflow in areas where that’s allowed (I’m describing myself here, in case you’re wondering)…. Sure, a normal person can do the JMT that early in a good snow year. The question then becomes, would I want to? That would be a big NO, because I would much prefer NOT dealing with long glissades down snowy passes wearing a heavy pack, ice and snow bridges, horribly cold snowmelt wading, and hordes of mosquitoes. (Old fogey status achieved.) I’ll wait until September when the chances of getting a permit increase, the crowds on the sections within a dayhike of a trailhead (first half and very end of the trail) decrease, and there are no mosquitoes to drive me into the tent in the evening.
I’m just trying to pack everything into the narrative that I see as a possibility, not be condescending, so I hope that my hasty post as I run out the door doesn’t misfire… Best I could do without a common idea of what normal is. That word makes less sense to me the older I get.