Well, the Only thing the C300 has going for it is its auto focus, which works better than most, if AF is what you need/want. Otherwise, it is an overpriced Prosummer Camera starting at $9995.00, with an EF Mount, so you are stuck with Canon AF lenses for the auto focus to work at its best, and a good cine zoom is going to cost you $10 more, and you still have a high a high,y compressed, low bitrate internal recording codec, AvCHD/H.264, which will work for streaming, but not much else.
If you want this type of camera, for less money you could get 2-p BM Broadcast cameras, have better lens selection; or get Ursa Mini Pros, and have a camera that can run circles around a C300, or get the new Panasonic EVA1, if you have EF lenses, or like them; but at least the EVA1can be converted easily to a PL mount.
I think yiur choice to go with the GH5S is a good one for what you eant to do —but, It all depends on what you want/need.