What are the main reasons for choosing a mirrorless camera with a viewfinder?
The main advantage of a viewfinder over using the back display is that in bright sunlight, it can be hard to see anything on the display. For example, I found it almost impossible to take pictures of birds in flight using the display because I could see hardly anything on it.
Another factor is that the viewfinder uses less power because it is smaller, so your battery will last longer than if you have the main display on all the time.
Mirrorless cameras largely provide an EVF for the same reason that DSLRs still have an OVF when there is a back panel LCD.
The EVF is smaller, can be dimmer and consumes less power, so there is a significant battery savings.
Holding the camera up to your face provides a third point of contact and makes use of your head's natural stabilization, which results in more stable images.
Looking through the viewfinder is a more immersive experience as the viewfinder fills your field of view and blocks out distractions.
The EVF doesn't have to compete with the outside lighting as external light is blocked out.
EVF can use diopter correction for people with glasses (though this isn't a huge deal since they could actually use glasses with a back panel).
There are some situations where an LCD may be preferable, such as shooting on a tripod or from an odd angle, but our eyes are the way we see the world, and shooting photos based on what we see rather than what we see on a small screen is simply a more natural experience, in addition to being a more power efficient and more stable one.
When you hold a camera at arm's length to frame a photo in the rear LCD, the camera isn't as stable as when it is held close to your body. There are basically two reasonable ways to do that: one, with a waist-level finder as on many medium-format cameras; or, two, with an eyepiece finder.
Since most small mirrorless cameras are not designed in a way that makes the waist-level idea practical, holding the camera to the eye it is. This also has some other property which can be an advantage: as human beings, of course, our view of the world is through our eyes, and putting the viewfinder there makes the shooting experience more immersive. The camera, in a sense, becomes an extension of your eye. This isn't always ideal — it's also often nice to have an awareness of context — but I find myself feeling a little frustrated when using a camera which doesn't offer me the option.
There are also some technical advantages, like better performance in bright daylight, and the lack of a distracting glow back on your face when shooting in low light, but overall I think the perspective issue is the most important.
And of course, this works the other way around, too: there are many situations where a tilt/twist rear LCD screen lets you put the camera in a place to record a perspective that works better than to your head. It's nice to have both possibilities available.