Lightrooms default sharpening is far too high for Fuji X-trans files, I have set up a custom import profile that detects Fuji X-Trans camera (from a list I've entered) and sets a lower level of default sharpening.
Lightrooms processing has improved considerably over the last few years for X-trans, but it still doesn't extract as much detail as other solutions. I'm now using DxO Pure Raw as a plug in for Lightroom for RAW processing.
Going back to the OP, early mirrorless cameras are nowhere near as good (especially in focus speed, EVF quality, LCD screen quality, object tracking etc) as later cameras. Also joystick access to the focus point makes it easier to focus correctly on the target area. Autofocus speed will matter when pgotographing puppies/dogs. Lens choice is critical as it is with all photography.
In your shoes I would suggest staying with what you have, the 6D was (and still is) a good camera (possibly lacking in focus points, but it does have a sensor that works well at higher ISO's). Save up, and whan you have a decent amount of moeny in the pot you can look at switching to a mirrorless solution.