Why not combine a wide angle lens with a second lens that captures a small portion of the wide angle lens's image? The wide angle lens would capture a large field of view and a large depth of field. Then, the second lens magnifies a small portion of the image formed by the wide angle lens.

That second lens is called a "teleconverter". A "2x" teleconverter has the following properties: it doubles the focal length compared to that of the unadorned lens. It doubles the aperture number (there goes your light). It doubles the size of any lens defects (fortunately the worst defects tend to be in the corners of the image which we throw away anyway, taking only a central crop), adding a few of its own. Since all the gathered light in the non-central portion of the original lens ends up inside the lens combination somewhere where we don't have 100% absorbent black available, we get some loss of saturation and contrast from the scattered light of the abandoned image parts.

What happens to the depth of field? It halves. After all the above sacrifices, particularly doubling the aperture number, why would the depth of field go down instead of up? Because the depth of field is defined as the displacement with regard to the focusing plane where the "circle of confusion" in the image reaches a certain size, and since the sizes in the image all have doubled while the entrance pupil size stayed the same (it is focal length divided by aperture number), the displacement of the scene where we reach the same circle of confusion size has halved.

So no, this kind of tomfoolery of thinking the lens into two separate independent parts buys us exactly nothing. There are some fancy names when such a design helps creating lenses that are smaller than they "should" be, like "retrofocal design", but it pays off to optimize the groups of a lens as a whole rather than individually.