The fundamental driver of cost in a lens is not the correction of aberrations, although the correction of aberrations does add to the cost of a lens, and may be a more significant factor in wider angle lenses. Generally speaking, the primary cost of a lens is the "glass". I put glass in quotes, because sometimes it is other materials, such as Fluorite or a diffraction grating or diffractive particle dispersion, however advanced lens elements usually cost MORE.
You cannot achieve a specified aperture without having the appropriate magnification from both ends of the lens for that aperture to appear to be the correct size. The notion of a "physical aperture" is generally a misnomer. What we call the aperture of a lens, what is frequently referred to as the physical aperture, is correctly termed the entrance pupil. The entrance pupil is the aperture as observed through the front of the lens at a distance of "infinity" (or, in other words, a sufficiently great distance that the observation is of collimated light.) The entrance pupil of a 600mm lens with an f/4 relative aperture must be 150mm as observed through the front of the lens. To achieve that magnification, two things must be:
- The right lens elements each with the right magnification must be used to achieve that magnification.
- The front element must be at least 150mm in diameter.
Think about this for a moment...a 150mm diameter front lens element. That is 6" in diameter, about a hands breadth. That is HUGE. On top of that, the front half of the barrel up to the diaphragm is only slightly tapered, and there are a number of additional lens elements that must be used in addition to the front element to achieve the point #1 as well as correct for aberrations. So you have a number of 4" to 6" lens element in the front half of the barrel, on top of all the lens elements behind the diaphragm to properly project a rectilinear image onto the sensor, each of which are still an inch to several inches in diameter. All of that glass COSTS!
Replacing an aspherical element that corrects for aberrations with a spherical element that does not will probably reduce cost, but not by a particularly significant amount unless we are talking about very short focal lengths are wide angle zooms (where the cost of aberration corrections tends to be a more significant cost factor, as overall glass quantity is much lower than in longer focal lengths.) Regardless of the type of lens, however, usually the larger portion of cost is the front element, maybe the front couple of elements.
Even in a wide angle lens, the front element will usually be many times larger than necessary to achieve the correct entrance pupil, only in this case it is necessary to bend light from a sufficiently wide angle, rather than to gather the required amount of light. In a wide angle lens, the front element can be many times the total volume of any other single lens element. Glass costs.
As you state, the amount of additional glass need only be enough to allow for the larger aperture. Remember that every stop in aperture is a FACTOR OF TWO change in aperture AREA. If you have a 600mm f/5.6 lens, the aperture is 107mm, or ~11,500mm^2. If you want to make a 600mm f/4 lens, even if you don't care about correcting aberrations, that is a 150mm aperture...or 22,500mm^2!! You've DOUBLED the minimum area required to support the aperture, and likely more than doubled the volume (larger elements are usually thicker as well, so the total increase in volume can more than double). And that is just for the front element...there are still around 12-18 more! The amount of glass, in terms of volume, required to just barely be enough to support the larger aperture is more than double what is necessary for the next stop down. Don't underestimate that cost.
As others have mentioned, a lot of lenses already do exactly what you have theorized: Allow IQ to suffer at maximum aperture, requiring the lens to be stopped down by as much as a stop or so to fully realize maximum sharpness potential. Generally speaking, cheaper consumer grade lenses do such, usually as a result of a multitude of factors (cheaper glass, simpler and fully automated manufacturing process, automated assembly, etc.)
The single primary reason why cheaper lenses are cheaper, however...is smaller maximum apertures. Most consumer-grade lenses, as well as most third-party lenses, use smaller maximum apertures. Most consumer-grade wide angle and telephoto zooms max out at f/3.5, usually non-constant so f/3.5-5.6. Many third-party telephoto zooms start at f/5.6 while brand names start at f/4, and third-party telephoto zooms will often use f/5.6-6.3 while brand-name telephoto zooms often offer f/4 or even f/2.8 constant aperture. Maximum aperture is the primary driver of cost, as it explicitly drives the total volume of glass required.
Corrective lens elements designed to reduce aberrations, such as aspheric elements, Fluorite elements, diffractive elements, ultra-low dispersion elements, etc. all add to cost, however again making a large 5" fluorite element for an f/4 telephoto lens is considerably more costly than making a 3" fluorite element for an f/5.6 telephoto lens. Again...every stop change in aperture is a factor of two change in area, and an even greater change in the total volume of glass.