Sony Alpha a6500

Pros

• I am still using an old Sony A6500 and no real complaints about it.
• Stepping up to A6500 and Sigma 60mm F2.8 and its itty bitty built in flash the pics are nice and sharp but its not exactly a small and light camera tho compared to A7 and enormous 28-70mm F2.8 it is :)
• See if you can find a Sony A7iii or a 6400/a6500 with your budget, it would be better for video.
• I have a sony a6500 which is smaller and lighter than all the cameras you mentioned except the gr
• I have an a6500, and it takes great photos. The only drawback is the screen can’t flip out if you want to vlog with it.

Cons

• Didn't test specifically on 6700, but tested on 6300 and 6500 and nitrate made no difference , recording or not, made no difference. You'd overheat as fast actually recording, as you would browsing the menus
• I am still using my A6500 tho.
• I do professional photography at parties, weddings etc with sony a6500 which has worse AF than a6400 and is older.
• The 6500 is just not a huge upgrade from the gh7. The 6700, for sure, but the 6500 is quite long in the tooth and not like it’s a full frame anyway.
• Cameras you're considering: Sony a6400, Sony a6500, Sony a7r ii