We love summer at Disney, have done late June and early July many, many times.
Crowds are typically lower than other times of year, so that's a huge plus. Yeah, it can be hot...but it's central Florida in summer, duh. Besides, it's not like you spend all day standing outside in the sun, unless that's what you choose to do. There are very simple ways to avoid the worst of the heat while still getting to do a lot. It's hyperbole to say "It'll rain every day". It may or may not. We've had trips where we got t-storms 6 or 7 days in a row. We've also had trips where we didn't get a single storm for 10 days (there were a few storms, but they happened in the middle of the night). So be prepared for storms, but don't think it's a guarantee they'll happen. We also just like the warm weather at Disney. In our opinion, a cold Disney isn't fun. We like being able to ride Splash, Kali and get wet. We love the water parks.
Honestly, I can't think of any true cons for us in late June. If you have a major adversity to heat, then you'll have to make sure you plan smart.