Film SLRs still used the 2 curtain method to open and close the shutter. This is not a rolling shutter. That 2 curtain/fabric shutter is not what causes the rolling shutter "effect". The rolling shutter effect that causes "slanted vertical objects when shooting out of a car window" is due the way the digital camera gathers information from the sensor. Digital CMOS sensors electronically scan the image line by line, from top to bottom. The amount of time that has passed between when the sensor captured the top of the image and when the sensor captured the bottom of the image is significant enough to some times make things look weird. This is what causes the slant-y lines and what causes the example at the end with the bottle of champaign.

This site has a good explanation of it in addition to some good illustrative videos: http://www.diyphotography.net/everything-you-wanted-to-know-about-rolling-shutter/

The guy in the video didn't explain it very well. A "rolling shutter" is how to describe the way the camera gathers info from the sensor. That does not change based on which mode you are in. What changes is whether the sensor acts as a 'electronic' sensor (like during video mode) or a mechanical sensor, (like during still photo mode).

Go to page 17 of this power point: https://www.smpte.org/sites/default/files/section-files/Walt%27s%20Rolling%20Shutter%20PowerPoint.pdf

and you will see a comparison between a global shutter (The CCD image on the left) and a rolling shutter (The CMOS sensor on the right). The CCD camera does not have the "rolling shutter effect" because it uses a global shutter. Film cameras also use a global shutter because they expose the whole film plane at the exact same time. A CCD camera sensor gathers all of the information from the sensor at the exact same time. A CMOS sensor (like in your phone and most DSLR cameras and the DSLR camera in the video) gathers information line by line. This is why the cork/champaign image happened the way it did.