I do image stacking all the time. Two main reasons, to get the equivalent of a longer exposure time (for smoother water/clouds) or to reduce noise (generally astro landscapes).
You can do this in PS with smart objects but it is slow. However there are other programs that can do this much faster such as Long Exposure Stacker, image magick, etc. Sony used to be able to do this in camera with the app Smooth Reflections.
Mean results in slightly softer images but can leave ghosts if objects move. Median tends to be crisper but eliminates objects that are only in one frame like a moving airplane, car, person, etc. Can sometimes result in weird artifacts in smooth areas though.
I often use this technique in conjunction with ND filters. For example, a 10-stop ND filter might give you a 10s exposure in daylight. So instead of getting a 16-stop filter, just average 10, 20, 40, etc shots with the 10-stop to get to minutes in daylight.
For astro-landscapes, I might take 5 2min exposures and average them to get lower noise on the foreground instead of one 10-minute shot. This is very helpful as some types of noise build up with longer exposures and occasionally shots get ruined because of headlights from passing cars. With the averaging case, you just throw out the one with the car and use the rest.
The main drawback is that if an object is moving fast, it can leave gaps between shots. E.g. A car trail split amongst two pictures will have a gap due to the delay from one shot to the next.