Could you elaborate on what's different between shooting with adapted lenses on the Sony and the Fuji? Does the Sony handle adapted lenses differently, or is it something else?
First off I need to emphasize that I have some very specific needs when it comes to this kind of thing. For instance I have a visual problem that makes it extraordinarily difficult to rely on an EVF for focusing. I'm sure under different circumstances this wouldn't be an issue for me. So here goes...
For me it centers around two things, focus peaking and the LCD.
First off focus peaking on the Fuji cameras is a lot less distinct than on the Sony cameras. It's a different approach and it can be argued that the Fuji is a bit finer than the Sony approach. It's also less visible. In practice, Sony's more coarse but VERY visible approach to focus peaking is easier to master and I am able to focus very quickly and get a high percentage of in-focus images. Sony's focus peaking even at the lowest setting is more visible than the Fuji's at their highest setting. It comes in three colors (red, white and yellow).
Next, I find that focusing legacy lenses (which is often lager and heavier native glass) benefits from shooting at chest or waist level. it improves stability. All Sony bodies have a gorgeous flip down LCD. MY favorite feature of that LCD however is a "Sunny Weather" setting that boosts the gain and makes it visible in all but direct sunlight. I do believe some of the newer Fuji bodies may have this now.
Compared this to my XE1 where the focus peaking is not very distinct, no flip down screen and a rather laggy EVF. OF course newer models like the XE2 have a significantly improved EVF and added red focus peaking. The XT1 has a huge and gorgeous EVF and flip down screen but the focus peaking is still not anywhere as pronounced as the Sony. That's not to say that I can't use adapted lenses on the XE1 (clearly I can) but it takes a good deal more effort. I recently borrowed a Sony a6000 from a friend and I was able to throw on any lens walk out the door and shoot quickly and get a high percentages of in-focus images.
For whatever reason making the Sony work well with adapted lenses was something their engineers put a good deal of effort into and it shows. If I was looking to shoot native lenses they certainly wouldn't be my first choice but for folks that like to go old school like me they are worth a look.
I am not saying that I think Sony is better than Fuji. I actually prefer the images the Fuji produces. Each has their strengths and weaknesses. It's up to each of us to figure out what is important to us choice the tool that best suits our needs.