I haven't seen as much from the Panasonic models, but it's been my experience Sony tends to yield a cooler image than other brands when in auto mode. In manual white balance mode, the difference is much less. Since I've shot professionally, I tend to shoot in manual mode even on consumer cameras.
The one pro Panasonic model I tested once back in the SD video days yielded similar results to what you described: flat colors with a more washed out appearance. Tried as I might, I couldn't get truly acceptable video with it for commercial production. We ended up sending it back and bought a nice Sony Hyper HAD shoulder cam with a Fujinon lens that produced tack sharp images even on analog Betacam. The best part is, we were able to add it into the studio setup for larger productions requiring more cameras since they were the exact camera with studio connections instead of the dockable deck. Being the same camera, we could just use the same scene file in video shading, with just a few minor tweaks.
The one thing I did like about the Panasonic was its low light capabilities. I guess the flatness was the compromise needed to get there. Had it been for news photography instead of commercial production, I might have OK'd the camera for all those times news crews are shooting indoors and at night.
I do prefer the saturation and black levels of Sony cameras, but they do require careful white balancing to avoid looking too cool. I've had mixed lighting conditions where the placement of the white card made a massive difference.
At one station I worked at years ago, the sports guy always tried to get me for his shoots because he liked the look of my video under stadium lights. Too many other photogs would come back with orangy video from the sodium vapor lights or blueish from mercury vapor ones. I would always try to get an opportunity to have someone hold my card out on the playing surface to ensure it was fully within the field lighting.
Later, in the Creative Services department, when I did commercial production, I often found myself in store front businesses with large windows, fluorescent lighting and lots of shadows. It took a lot of finessing to make those places look respectable with a small Omni Light kit to even things out.
The common thing between both situations was a lack of advance post production correction equipment like we have today in digital editing systems. I had Video, Setup, Chroma and Hue adjustments and a scope to monitor them. That's it. Now, everything is pixel-based, so you can control not just individual colors (and even correct white balance after the fact) and gamma levels, but individual parts of the image. It used to be all or nothing. Now, you can bring out one little detail while leaving everything else intact. I get some really awful video at times and it's amazing how good I can make it with the capabilities of today's non-linear editing systems. Add in tools like After Effects and control is endless.
Having said that, there's still no substitute for properly shot video. The less tweaking I need to do, the better for image quality. Too often, the more you have to do to make video look acceptable, the more compromises that occur compared to the same video shot properly.
At any rate, I opted to go with the AX100. I seldom shoot hand held, so I can do without the BOSS stabilizer. I've only had it for the day, but I'm already very happy with the results.