The biggest reason many folks who use an f/2.8 lens for landscapes shot at f/8 do so is because they already have an f/2.8 lens anyway for when they need f/2.8 in other shooting scenarios.
It is generally true (though there are always exceptions) that f/2.8 zoom lenses are built more robustly than narrower, variable aperture lenses. Thus they can withstand a bit more abuse in the field and will hold up to steady daily use over the years.
It's also generally true (though there are always exceptions) that f/2.8 zooms tend to be sharper, even when stopped down, than their variable aperture counterparts. The difference by f/8, though, is usually fairly negligible.
The biggest differentiator between most "pro" landscape photographers and their "amateur" counterparts is technique while shooting and post-processing skill.
There are, however, more than a few specific examples where a narrower aperture zoom is sharper at common focal lengths and apertures than a more expensive wider aperture counterpart from the same manufacturer.
Take, for example, the original Canon EF 24-70mm f/2.8L. Though it is plenty sharp, it's not quite as sharp at many common focal lengths and apertures as the newer EF 24-70mm f/4L IS. Even the newer EF 24-70mm f/2.8L II is not noticeably sharper at f/5.6 and f/8 than the f/4 that costs $1,000 less. The f/4 lens is also lighter, smaller, cheaper than both the original f/2.8L was when it was in the catalog and the current f/2.8L II, has a shorter minimum focus distance for higher maximum magnification, and has Image Stabilization that both of the 24-70mm f/2.8 lenses lack.
Another example would be comparing the EF 16-35mm f/4L IS to the EF 16-35mm f/2.8L III. Again, the f/4 IS lens is smaller, lighter, cheaper, has IS, and costs about $1,100 less than the f/2.8 III. Yet the $1,000 f/4 is as sharp as or sharper than the $2,100 f/2.8 at all focal lengths and common apertures other than at 16mm, where the f/4 is sharper in the center but the f/2.8 III is slightly sharper on the edges and in the corners.