When taking a still you have a physical shutter which controls light hitting the sensor. The image is still read line by line when taking stills but because of the physical shutter closing no extra light is recorded during readout.

The line by line readout is a consequence of the CMOS design found in large sensor cameras, and thus is unavoidable (until they find a way to make cheaper/bigger CCDs). 

It's worth noting that at shutter speeds faster than the cameras x-sync speed (usually 1/250s) the shutter starts closing at the bottom before it's fully open at the top. The result of this is that for really fast speeds like 1/4000s you get a slit that traversed the frame and gives you a similar rolling shutter effect for stills. However because the time to traverse the frame by the shutter (1/250s) is ten times faster than the time to read the sensor during video (1/25s) you need a really fast moving object to notice it. 

Here's a very old photograph which demonstrates the effect well: 

<img src="https://4.bp.blogspot.com/_ZoApFAh9Orc/TDi49eFQ9RI/AAAAAAAAAZk/3Y8fy3DGQ1w/s1600/HENRY_LARTIGUE_aberration_RACE+CAR.jpg">

The effect is also more noticable in videos if you pan back and forth, which doesn't happen with stills.