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I'm still using my 18-105 kit lens on my Nikon D5300, and the (lower) quality of the photo on edges is bothering me, so I want to get some lens that produces good quality through the whole photo. Still deciding between prime vs zoom for quality vs multi purpose use, but what I'm really not sure is if/when low F-stop is really important for such photography.
So why should I consider buying a faster lens since all being in focus is one of the most important things when doing landscapes?
When used at the same aperture and focal length, a faster lens will have less vignetting than a slower lens.
For example if you choose to shoot at 35mm f/4.0, then a 24-70mm f/4.0 lens will have the most vignetting, a 24-70mm f/2.8 lens will have less, and a 35mm f/1.4 prime will have the least vignetting.
The reason this happens is because of tradeoffs in lens design. Vignetting occurs because the front element is not big enough to match the aperture. Making a vignette-free lens would require very wide front elements, which increases weight and cost.
Vignetting test samples can be found on The Digital Picture.
One reason to avoid vignetting is for panorama stitching - the brightness must be made consistent before stitching can begin. Another reason is if you want to start with a vignette-free image and deliberately add vignetting in postprocessing, but in a manner under your control.
A faster lens (= low F-stop) allows easier manual focus and better auto-focus, given the camera has AF sensors which can exploit faster lenses. For landscape however, there is usually enough time that there is not much to gain from this.
Faster lenses often surpass slower lenses when stopped down to the same F-Stop (eg. f/8). The reason is not that a faster lens is better by default. Rather it comes from the required optical quality for a fast lens, which should be useable when shot wide open. Also a faster lens, especially when it comes to zoom lenses, represents increased cost in manufacturing due the larger size of the lens. Thus they often will be designed to be the top of the line products of each manufacturer.
It is also possible to build "not so fast" lenses with comparable optical quality, leading to a more compact lens. Many Pentax prime lenses represent this approach.
Consider low light conditions. If you can use a low F-stop, you can use a lower ISO. You'll then have less noise and higher dynamic range. Even with a tripod, you'll have more options if you only need an exposure time of, say, 0.3 seconds instead of 2 seconds. In very windy conditions the tripod may shake, but even in alight breeze tree branches may move a bit.
The maximum sharpness reached at the higher F-stop where you are just above the point where you get diffraction, usually around F/8, will be much better. So, pictures in broad daylight conditions will also improve. You can also consider using a zoom lens and composing a high resolution by stitching a panorama. Pictures taken using such a zoom lens will be less sharp per image, but each image is then part of a larger panorama, when that panorama is resized to the same format as a single image you can take with a, say, 50 mm prime lens, the result will much be better. But, this method requires a static scene, so it's not always applicable.
Faster lenses give you almost always better image quality than slower lenses at the same f-stop. This is true for vignetting, resolution and contrast, distortion and color.
A faster lens also gives you a brighter look through your viewfinder or less noise on your live view display. Anyone who has tried to focus on a star at night knows how annoying a noisy display can be.
Also, fast lenses are usually more expensive and often built better. One extraordinary example for this is Canon's EF 35/1.4 II, which is very durable and is weather sealed. Other brands have well built and sealed lenses, too.
I see no advantage regarding auto focus though, as for one thing no camera I know uses sensors that would benefit from an aperture opened wider than f/2.8 and because modern cameras can focus with as little ambient light as LV -3.
Lastly, if you need or want to take a picture with a wide aperture, you cannot open a f/4 lens to f/1.4, while you can always stop down a f/1.4 to f/4. The fast lens gives you greater flexibility.
There are also reasons to not spend the money on super fast lenses:
They weigh more, cost more and may even require more expensive gear, like a tripod and head that can carry more.
They need more room in your camera bag, leaving less for possibly important stuff like graduated neutral density filters, which often help getting better landscape pictures.
In general, at wide open apertures they are prone to purple fringing and focus shift (this latter problem should noth bother you when landscape shooting).
(Keep in mind, that there are many great and famous landscape pictures that were taken many years ago, using films, cameras, and lenses that today's amateur photographers wouldn't even want to touch.)
Faster lenses aren't necessarily inherently better for landscape photography, because the implication with 'landscape' is a stopped-down shot, preferably on a tripod. Are you stopping down now? If not, it might explain the edge softness.
Low-light stuff or razor thin focussed portraits and flower shots etc obviously benefit from a large maximum aperture as more brightness and a narrower depth of field are needed for critical focus and the look of the shot.
Fast lenses tend to be the premium grade lenses, so they may be a little better optically. Typically, a landscape shot is about detail, so this is useful, albeit usually the differences are usually a lot less by f8 than at large apertures.
The equivalent aperture on a fast lens is more stopped down (e.g. f2.8 is wide open on a f2.8 zoom lens, but is two stops down on a f1.4 lens). Stopping down nearly always improves the image from wide open, especially in wide angle lenses, but the effect is reduced by f8 (even the f2.8 lens is closed down 3 stops at f8).
However, the optical difference is only one thing. Using a good tripod well is useful. A really poor tripod is close to useless, and too much movement and vibration will blur the pictures in even the most expensive lens.
Fast lenses are heavy lenses. This can be a disadvantage in landscape photography if they have to be lumped up a lot of hills!
Of course, that is all for a classic landscape shot and many do things differently to be impressionistic. In the end, technique can be as important as equipment, once a basic level is met.
