12_35 does work well. But for run & gun stuff a handheld stabilizer is needed. Shoulder rigs can work but small handheld stabilizers while expensive can be close to perfect. For other types of filming a tripod with fluid head is best.
I don't have one, but this technology looks very promising. Basically you'r trading cropping in post for what looks like rock solid stabilization. SteadXP - The Future of Video Stabilization
Here is a good article on the subject of stabilizers From what I've read there are serious technical issues for SteadXP for commercial grade work.
I use one of these for handhelp stuff, where the EM Mk II IBIS is not going to cut it or for smaller unstabilized cameras: https://www.amazon.com/gp/product/B007P8AXWK/ref=od_aui_detailpages01?ie=UTF8&psc=1 I also have a dolly/slider tracking shots. Can;t remember the name of it right now, but it is awesome because it is adjustable to use on rails, 2x4, 2x10, so is very versatile. It can also be flipped upsidedown and used on the ground or table top if need be. I'll look up the name later and add to this.
One of the most impressive portable setups I've yet seen is from Chris Prouse. She uses a 3-axis gimbal, a GX85, and the new 12-60mm for Dual IS. Obviously the Nokton is not readily compatible with a motorized gimbal, since you cannot manually focus it while it's in use...
The Flycam Nano DSLR is not really suitable for real run and gun. Try the DJI Ronin-M 3-Axis Handheld Gimbal Stabilize or something better.
perfect stabilisation costs money, and even then , learning to drive the system still requires skill and practice. I skimmed through some of your shots, and overall I thought you did a good job, you may just need to practice. Sadly the lighter weight of your camera negates some of the benefits of your rig, a passive system, which depend on mass and inertia to achieve the best results in smoothing out the camera movements. I also thought i saw in some of your outdoor shots a certain jittery feel to some of the shots. This may be due to using too high of a shutter speed check this out to understand a little more How Shutter Speeds & Frame Rates Can Be Used to Change the Look & Feel of Your Film cheers K
The one thing about video is that it takes a lot of practice even with expensive tools. I find that the 12-35 shooting 1080P on the GX8 where I can use the software/lens IS is ok and I can clean it up in Mercalli 4.0 where it "works" but I am now tending to shoot in 4K with lens IS, use pan and crop for movement. If I have to walk (old, no run & gun for me), Mercalli Pro 4.0 does a good job of smoothing things out. How many hours do you have in learning your FlyCam? Over on DVX I think someone said is was only after 100 hours that he could get reasonable footage at a slow walking pace - again that lots of practice thing.