Unngh.. howabout "Show, don't tell"? Text + still graphics = a webpage, not a film.
Seems like a key question for you is whether you plan on using AF during your videos on the A6300. With it's impressive AF system it might be worth having, and thus tip the balance towards buying native lenses. If not, and if size / weight is also not a big concern, then EF lenses are certainly more versatile for future systems.
One of the newer 3-axis gimbals combined with a lightweight camera should be pretty usable I would think. Not quite in the $300 range but the Zhiyun Crane is around $600 online.
Weight would be something like:
Zhiyun Crane 3-Axis Gimbal (1000g)https://www.amazon.com/Official-Zhiyun-Brushless-Handheld-Mirrorless/dp/...
Panasonic 12-35mm f/2.8 (305g)
Panasonic GH4 (560g)
For a total of 1856g / 4.1lbs
Could maybe save half a pound by swapping the camera & lens for a lighter combo like the GX85 Jeffrey mentions above (at the expense of some image quality perhaps?)
I've been using my Pilotfly H2 (another fairly new 3-axis gimbal but slightly heavier than the Crane) with a Sony A7R2 + various prime lenses and I've found one of the keys to reducing fatigue is to switch to inverted mode (upside down, holding it like a walking stick) which requires a lot less wrist strength. Obviously the perspective is much lower (chest height) but that can be good for a variety of shot types (e.g. pans of people sitting at tables, low-angle shots, detail shots, etc.)
The Pilotfly H2 also has a two-hand holder attachment you can use to get even more stability and spread the weight across both arms.
1200g / 2.6lbs
480g / 1.05 lbs