Let's look at it in terms of depth of field.
DOF Master gives for Canon 5D (a full frame camera), 16mm focal length and f/2.8 a hyperfocal distance of 3.03 m. If you focus at 3.03 m, everything from half the hyperfocal distance (1.52 m) to infinity is in focus.¹ Do you have elements in landscape pictures that are closer than 1.52 m away? Probably not.
The f/2.8 is more expensive simply because it has more glass. It is heavier too. What you gain in speed is lost in lack of image stabilization. Although depth of field at f/4 is better, it is good enough anyway.
So, do you find handheld shooting important? If you don't wish to carry a tripod, choose the cheaper f/4 with image stabilization. Do you take landscape pictures in limited sunlight with a tripod? You may find the f/2.8 beneficial.
I would say where the f/2.8 excels is astrophotography, where you may have a combination of landscape and stars, or only stars. You cannot use long enough exposures when taking pictures containing stars because the earth is continuously rotating, and therefore, stars would become short star trails.
I would pick the f/2.8 if I had about $2000 to invest in a lens, because I want the possibility to take pictures containing stars not becoming short star trails. Or actually no, I wouldn't pick Canon, I would pick a third party manual focus lens that is even faster! Something like f/1.4 would be ideal for astrophotography.
So, as summary, speed may be important (e.g. astrophotography), but the f/2.8 is not THAT fast, so it sits between fast and slow lenses. Not being fast enough for astrophotography, it won't be chosen by many, and not having image stabilization and being too heavy and expensive, it won't be chosen by many others.
¹ Note that hyperfocal distance is a property of depth of field, and thus changes for the same image with different display sizes and viewing distances. If a DoF calculator does not indicate display size and viewing distance, it is probably safe to assume it is based on standard viewing conditions: an 8x10 inch size viewed from about 10-12 inches. If an image shot at "the" hyperfocal distance calculated for 8x10" is displayed at 16x20", the DoF will not reach all the way to infinity. There are DoF calculators that do allow designating intended display size and viewing distance. This one from Cambridge In Color has a 'show advanced' button to access the feature.