Hi Richard,
The problem is not just shutter lag, there is "EVF lag" too.
By this I mean that there is a delay between the sensor capturing an event, writing it to memeory and displaying it on the LCD (be that a big one on the back or a non-optical viewfinder (the EVF).
EVF lag adds to your reaction time and shutter lag before the image is actually captured.
I have found this with any bridge camera and (I guess) it must also afflict the mirrorless models too.
Some modern bridge and P&S cameras have 'precapture' modes, but these usually come with limitations I personally cannot live with for what I shoot (long distance wildlife in continuous mode).
Therefore, if I can, I frame up, hold camera steady and watch 'real life' over the top of the camera (or with the other eye) and shoot when the action starts. This takes EVF lag out and just leaves human finger reaction time and shutter lag. The latter can be significantly reduced by having the shutter half presssed, so that metering, focusing and IS/VR stabilisation have already been done. That just leaves finger cramp to contend with
When a bird unexpectedly launches into the air from a perch, I find this makes a difference between getting several empty frames and getting the first frame with the bird's tail in

The knack then is to study your subject and recognise when they might launch to get them
in the frame next time (and pan in the direction of flight by instinct alone).
My point is that EVF delay does exist, and is significant, but isn't too easily measured, nor is it given as a spec. - I suspect all cameras will be similar, set by the frame rate) - just one more tip to improve your odds at capturing the moment.
Cheers,