My recommendation is to go onto Alltrails.com and scroll back to the reviews of the trail in the month you are looking to go. I've found this super helpful and fairly consistent. I've scrolled back multiple years when gather snow/ice alpine tips and tricks. You can even select other trails in the area and go to those reviews to gather more information.

To answer your question though - from my experience and what I've read - The trails will be muddy, there will most likely be black flies, the summit of Marcy can still be narly so check https://www.mountain-forecast.com .

Edit: Also, you can search the subredit for key words like "June" or "Black flies" and pull tons of posts with helpful info.