Maybe the Sony 18-200mm f3.5-6.3 OSS is the one you are looking for.

Here is a list of lens for a6300 and their purposes.

1) Sony 18-105mm f4 OSS for travel or portrait. Weight about 0.43 kg
2) Sony 10-18mm f4 OSS for wide angle. Weight about 0.22 kg
3) Sony 35mm f1.8 for portrait. Weight about 0.16 kg
4) Sony 55-210mm f4.5-.63 OSS for telephoto. Weight about 0.35 kg
5) Sigma 19mm f2.8 for wide angle. Weight 0.14 kg
6) Song 18-200mm f3.5-6.3 OSS all-in-one multipurpose. Weight 0.52 kg
7) Sony 50mm f1.8 for portrait. Weight 0.2 kg
8) Sigma 60mm portrait lens. Weight 0.18 kg

I personally not much in video recording so can't tell much about it.

But for video recording, I guess you will need a lens equip with image stabilization system, quiet motor and smooth turning focus rings which make the recording a breeze.

Perhaps this Sony PZ 18-200mm f/3.5-6.3 OSS could be an ideal choice.

Maybe other users can share their thoughts too.

By the way, if you just consider the video resolution then it is the camera body ability not the lens anymore.