We went in mid June with kids last summer. Rome - Venice - Amalfi/Positano. It was crazy amount of people but still not claustrophobic although maybe Venice kinda was

and not scorching heat by any means.
I would never ever go after June. Sorry, just being honest. I think June works in a lot of places incl Asia even though not ideal but into July is chaos.
But that's everywhere - we did Jackson Hole in Aug and OMG. We couldn't hike till dusk/sunset it was so crowded!!!!!
I think you should go but really manage your expectations and also try to stay off the beaten path. I didn't even know we were at Trevi in Rome as people covered it up there were so many!!
I think the weather will continue to be horrible in summer and crowds will continue even yr round so see it now if you have a chance as it'll be worse in future years
