Does shooting landscape at lower f-stop produce reasonably sharp images?
Even at very wide apertures when the nearest part of the scene within the field of view is several hundred feet away, as is the case in your example photo, focusing at or beyond the hyperfocal distance can be used to keep everything in acceptable focus. If the nearest object in the scene is further away than half the hyperfocal distance then you would focus somewhere just short of twice the distance to the nearest object.
Will the lack of sharpness become evident for a full-size, high resolution image?
Just because the focal length, aperture used, and the shooting distance may allow for most everything to be within the limits of acceptable focus doesn't guarantee a sharp image will be the result. In such a case as described in the question the largest contributing factor to blur is likely not the size of the circle of confusion or the resulting depth of field or even failing to use the lens' "sweet spot" at all. Rather it is probably the amount of camera movement allowed by the photographer holding the camera by hand.
So in the case of the article cited in the question the author is balancing the need for a short enough shutter speed to prevent unacceptable camera motion blur with the need for an aperture that allows for acceptable sharpness.
Look at it this way: You can have a 50mm f/1.4 lens that is theoretically perfect at f/8 and a film that is a theoretically perfect recording medium at ISO 25 but if you're handholding the camera and only have enough light to use a shutter time of 1/4 second with that combination you're not likely to wind up with an image that is acceptably sharp by anyone's definition. If you use f/2 and 1/60 second you'll almost certainly wind up with a much sharper image that might be acceptably sharp to you.
Since more detailed information is not given, we'll have to guess at a few things. Let's assume the photo above was captured on 35mm film by a 50mm f/1.4 lens. If the photographer used f/2 that means the lens was stopped down a full stop from wide open, yet it still allowed a shutter time 1/4 as long as an f/4 lens used wide open would have required and a shutter time 1/16 as long as an aperture setting of f/8 would have required for the same aperture and film speed and lighting conditions.
At 50mm and f/2 using 35mm film to produce an 8x10 print viewed at 10 inches by a person with 20/20 vision (which gives us the CoC of 0.03mm) the hyperfocal distance is 136 feet. That means everything from 68 feet to infinity would be acceptably sharp. If the nearest point in the vista is further than 68 feet, then you would focus at slightly less than twice the distance to the nearest object appearing in the frame to maximize the depth of field of the entire vista.
