I haven't used one lately, but ten years ago the delay between seeing the image and capturing it was so great that it made them useless for people photography (unless people are posing, but I like to capture real emotions as they are happening).
Are they better now? In my experience, the shutter lag must be close to 0 to be able to capture the right facial expression. Lag ruins pictures.
[1]https://www.dpreview.com/forums/thread/4703538
The downsides are price and battery life, and if you have existing lenses you’ll need an adapter..
DSLRs are worse for capturing the right facial expression in some ways too, because the mirror flip temporarily blocks the viewfinder. With an electronic shutter enabled, a wedding photographer on mirrorless will never have the viewfinder blackout during burst shots when trying to capture "the moment". The mirror/shutterbox puts the "real" world lag on many DSLRs in the 60-120ms range, which is not so hard for an electronic viewfinder to get competitive with.
Mirrorless models with electronic shutters in theory also have the option to continually "precapture" and then use an image from say 100ms ago when button pressed - the iPhone does this to reduce lag time even more, and try to eliminate human response time from the process. A DSLR can only do this concept if the mirror was up blocking the view finder.
While people assume that there is a degree of "WYSIWYG" with the mirror, this isn't always 100% true either. Cheaper/small prisms or pentamirrors result in much dimmer and smaller viewfinders than a good electronic viewfinder, especially on APS-C sized sensors, and 100 percent view coverage is rare outside of high end ones. AFAIK all electronic viewfinders to date are generally 100 percent coverage, and often bigger.
One enormous benefit for wedding/event photographers that mirror elimination accomplishes is making the camera silent. A high end DSLR operating at its highest burst speed is very noisy as the mirror slaps back and forth, which genuinely can be problematic in quiet or intimate settings. The mirror boxes also generally are only good for typically 100-150k activations, which lasts a long time, but they can wear out.
The newer ones like the Canon R3 are even faster and the Sony A1 can do 240fps in the EVF.
Another perk of EVFs: they're much brighter indoors as they simulate the actual exposure.
I'm very sensitive to lag to the point where I couldn't play Smash Bros on a friend's TV that he insisted was fine. I'm not sure what the actual numbers are but the EVF latency is for all practical purposes imperceptible, and the cameras can shoot much faster bursts to boot.
Plus, people tend to change their expressions when a shutter/mirror is clacking away and mirrorless cameras can be silent.