Why don't plane wings start at the front of a plane and end at the back?
You are right, a short wing span is a blessing for the structural design and the wing will be nice and light. But since you need a certain wing area to keep the aircraft in the air, the wing will have a long chord as well, and that won't be good for aerodynamic resistance.

For subsonic aircraft, you want the longest, narrowest wing possible: the lift create by the wing is caused by air accelerating downwards. The amount of air streaming in is a function of the span, so a long span accelerates a lot of air downwards only a little bit; a short span wing has to accelerate much less air much more. That creates induced drag: the horizontal component of the backwards tilted lift vector.

 

Another source of drag is the tip vortex, where air 'flips over' from high pressure below to low pressure on top of the wing. So for subsonic aircraft, the longest, narrowest wing is the most beneficial from an aerodynamics standpoint. It will add constructional weight and therefore increase weight induced drag, but will compensate for it by a reduction in aerodynamic induced drag.

Supersonic aircraft have an additional source of drag: shock waves. The air gets no warning that an aircraft appraches. The nose arrives first, and a shock wave streams off of the nose in the form of a cone. Inside the cone, the air has lower velocity - if the wing can stay within the cone, it creates much less supersonic drag.

Image source

The higher the supersonic speed, the more the shock cone bends backwards, and the widest bit of the wing needs to be more towards aft to remain within it. So for supersonic aircraft, the delta wings that you draw are in use. They create a relatively huge drag when flying subsonic though.