A family member has a Fuji, and ditto on the above message for a good raw converter. If I am wrong, would love to be set right by somebody.
Overall, in good hands, the new mirrorless systems are amazing. I have seen billboards where the photograph was taken by a mirrorless camera and it was not full frame sensor for sure on that camera. The skill of the photographer cannot be replaced by technology, IMO.
Why I do not use one, and why I do not like them boils down to asthetics and a few other features. Your mileage can and will vary.
First off, I have an eye problem - extreme sensitivity to bright light. I have to wear sunglasses inside under some artificial lighting for example. For the life of me, even trying several different types of glass and filters on the eye side, I just cannot take to the electronic viewfinder. I much prefer the "real" sight/viewfinder of a SLR, digital or film. Also, for what it is worth, I seldom use live view on any of my DSLRs too. But that is a medical issue with just me and maybe not too many other people.
The second issue is battery life. ON all cameras, DSLR to several compact systems, auto focus, IS or VR, GPS, bluetooth and/or wifi, etc, etc, all chew up battery power pretty fast. For me, and again, your experience will differ, I find using manual focus lenses (mostly Zeiss) on my D700 and D800, with almost everything else turned off, really gives me a long battery life for shooting. ON a proportional basis, I find on the few I have used or tried, batteries can be depleted pretty fast on some of these compact systems.
But the bottom line is better to have any camera than no camera, and if you do not or cannot carry around a full sized DSLR, then these compact systems can be a great thing to have. Use the tools you need to get the job done and pay the bills is what it is all about.