I make things
I shoot (now) on dedicated video cameras, but I have shot with numerous Sony cameras and the GH4/5. Since I am also a photographer, I would personally get the a7 III, but if video were my primary or only concern, it would definitely be the GH5 (or GH5s, more likely) with a speedbooster.
The only real advantages I see the Sony having are a 6K down-sample and of course the full frame. Neither of these things is really a big deal to me - chasing the extra sharpness that a 6K down sample gives you is silly to me, and I'm much more accustomed to Super 35 size sensors (which a GH5 + Speedbooster will pretty much accomplish).
The a7 III probably has an advantage in low-light, but keep in mind if you're using a speedbooster you do get a "boost" in that regard as well, and (though I don't know for sure) that probably closes the gap for the most part.
GH5 does, however, have these far more important qualities: 10 bit 4:2:2 internally, better color science (I am not a fan of Sony's CS in regards to video), better IBIS, and a better form factor for video. The a7 III probably has a *bit* of a DR advantage, but once again, I think the GH5 closes (or exceeds) that gap with its higher bit rate 4:2:2 and therefore extra post-processing latitude. The GH5s also has dual native ISO.
But, personally, if video were my only concern, I would skip both of those entirely and get the new Blackmagic 4K Pocket in a few months, with a speedbooster, and have a little money left over. Internal 12 bit RAW, 10 bit ProRes, excellent BlackMagic color science, dedicated video controls, dual native ISO, less money. Bonus: same mount as your GH4.
I think some of these designs are really cool, and I might even buy some of these. But I'd be even more inclined to buy some if they didn't literally look like a rectangular picture cut out stuck onto a shirt. I wish the designs blended more (or at all).
Stopped watching as soon as he kept referring to 16:9 as "full frame" and then something about imax that made no sense.
I have a Nikon G for my BMPCC. Nikon glass is (and always has been) truly great. Go for that.
Plus, I do believe it's a few hundred cheaper than the Canon EF speedbooster.
Also, ignore Stephen. Cameras like the A7s are great for many things - video included - as well as, of course, photography (the main reason they were made), but in pure image quality (and not battery life or all this other stuff that in the end doesn't show up on screen), the more affordable BMPCC blasts them out of the water.
There are many people who are bitter toward Blackmagic because, admittedly, Blackmagic has issues with things (like battery life, ergonomics, etc.). But, as one of the fellows who runs BM said "what we try to do is deliver a lot of camera for a little money. and we deliver a whole lot of camera."
Sometimes that comes at the expense of some other things. It's like that old saying of fast, cheap, and quality but you can only have two. BM chose cheap and quality because that's their market.
You can complain about a lot of issues with the BMPCC (or BMCC). Battery, poor screen especially in day light, not the most efficient controls, but image quality?
That's just a joke.
If you think there is any camera on the market that can produce equivalent image quality for the same price-point, you are fooling yourself. And you can talk about all the accessories and whatnot you need for it, but I've gone out with just a BMPCC, a lens, an SD card, and a tripod, and made footage that would make someone with GH3 weep. Sure, it was tough to focus in the daylight, but I'd put a black towel over my head to cut the light, and it wasn't so difficult.
I paid $800 for my BMPCC which included tons of accessories.
Literally a joke if you don't understand why they are so highly regarded for their image quality at that price point. "just a run of the mill mirrorless camera with RAW" is naive.
I use a Nikon speedbooster with my BMPCC, but before I got that I had both a Panasonic 12-35 and an Olympus 12-40 (not at the same time obviously, as they're very similar lenses). I much prefer the Olympus - not only do you get the meager extra zoom, but it has a repeatable focus rack, which is lacking on the Panasonic.
If I had my druthers I'd get a Minolta MD speedbooster for it (which Metabones does not make - they only make one for Micro 4/3), but there are a few off-brand ones. I have no idea how well they work, I've never used one, but I'm always inclined to try them. I have a lot of nice Minolta glass and love being able to control aperture manually.
But, if you are looking for a straight lens with no adapter or Speedbooster, I recommend the Oly 12-40 f2.8. Served me well for a long time.