Given the fact that on a mirror less system, you have to focus and shoot with the LED screen, one would expect shutter lag and focus being an issue but it works very well.
This is somewhat of a misnomer. On a DSLR (and late-generation film SLR) cameras, you have a dedicated phase-detection array behind a pellicle mirror. With phase detection, the camera can sense how far out of focus it is on any given point, and drive the lens directly to the point where it will be in focus. If, however, you want to frame using the LCD screen rather than the optical viewfinder, that pellicle mirror has to swing out of the way, and the phase detection array isn't working anymore, so the camera needs to focus by contrast detection - driving the lens back and forth until it can find areas in the image where you have a sharp transition between different colors, and when that transition is sharp, it means that the lens is focused. This is slower and less reliable than phase detection, but lacking a mirror, this is the method that most early mirrorless cameras and virtually all fixed-lens compacts use, leading to the shutter lag they're known for.
However, most modern mirrorless cameras (almost all the Sony Alphas, Fujifilm X-series, Olympus E-M1 II, Canon M and R series, Nikon Z series, some Leicas too, I think) as well as some compacts (Sony RX100M5 and up and I think some Fujifilm as well) use hybrid autofocus where the phase detection points are integrated into the camera sensor, making turning lens hunting and shutter lag into things of the past.
Notably, Panasonic is sticking with contrast-detect autofocus even in their high-end offerings like S1R (they have a proprietary system they call Depth from Defocus, which claims to interpret an unfocused image to guess how much out of focus it is and in which direction, but by all accounts it doesn't work as well as true PDAF systems), as does Olympus on cameras below their flagship EM1 II and EM1 X offerings.
As already pointed out by another poster, on mirrorless cameras you can use the viewfinder alongside the display (assuming you have one - some, like Sony A5100, skip it), but unlike SLRs, it has no effect on focusing speed and accuracy. It does give you some advantages over an optical viewfinder such as the ability to zoom in and see fine detail while focusing, but overall it's a matter of personal preference. I dive with a Sony A6300, which has an EVF, but underwater I prefer using the screen - peeking into the viewfinder through the mask is just way awkward. I suppose add-on viewfinders mitigate this somewhat, but they tend to cost around a thousand dollars apiece, and mount to housings that cost several thousands, whereas my housing cost me $500 shipped with two ports (flat and 6" dome).
