First and foremost, without a mechanical shutter of some kind, there is no way to actually stop light from reaching the sensor after the exposure time has passed. With a purely electronic shutter, you continue to expose as you read out the image. Readout is fairly fast, but outside of specific cameras that use a high speed global shutter, often not fast enough to avoid exposure quirks and other artifacts.
With cameras that use a rolling electronic shutter, whenever there is motion that moves at a higher frequency than the shutter itself, you can get stuttering or jittery motion for the object(s) that are moving across the frame. The same wobbling that occurs with rolling shutters during video can result in torn or warped frames, especially if the photographer stops "holding still" while the exposure is still being read (and yes, this can still occur in the fraction of a second it takes to read out the whole sensor.)
The only way to GUARANTEE that your exposure time is 100% accurate, and avoid any unsightly artifacts caused by continued exposure during readout, is to use a mechanical shutter. Focal plane shutters are the most common, but leaf shutters in the lens would serve the same purpose with an electronic shutter. Anything that entirely blocks out light is necessary to ensure exposure functions properly. Modern global shutters with a very high speed readout are capable of replicating mechanical focal plane shutter behavior, however global shutters are not very common on still photography cameras yet. They are mostly used for video cameras, and the good video sensors with high speed global shutter tend to be very expensive.
There may come a time in the not too distant future where a good high speed global shutter that can read out the entire sensor in a small enough fraction of a second that they could effectively replace mechanical focal plane shutters. Until that time, mechanical shutters solve some critical problems, and are essential.