Well then why not cut the wings in half and lighten the airframe? Because you would then have to land at a faster speed. Not a great trade off.

Tell that to all the owners of clipped wing Cubs.
The purpose of anhedral and dihedral is to deal with stability, not to make the plane land faster or slower, to lighten the airframe, or to make it stronger. As said earlier a by-product is slightly less tension on the wing as Shaun tried to allude to.
It's not wocket science... but it could be

Well, the wing is not in tension. Its in bending, the by product of which is tension on one side (normally the bottom in straight & level flight) and compression on the other. Clipped wing cubs land slow enough that it don't matter, not so with most jets!

Jets land so fast it don't matter.

Interesting topic, I learn something new everyday. I have been wondering what a swept back wing with anhedral would do for stall characteristics on an airplane such as a Gnat?

I found a picture of a Gnat that shows the anhedral angle a little bit, here it is.

If the tips were constrained like the ends a bridge, but they aren't. The mechanical structure of the wing is what keeps the wings in place, not the geometery that is in question here (3 - 5 degrees of anhedral). You could only play this if (in the case of a B-52) the wings might 'dihedral' out of spec it flight, then drop a little extra anhedral to counter the distortion of the wing structure - both wings would support the same weight though.

Lift increases exponentially to airspeed, so you would need so little airspeed to make up the topview platform wing area difference that it would never be measured. You are also losing area at the tip which is negligible. This is also ignoring any lift provided by the fuselage. Looking from the top, the percieved wing area would be nearly identical in both cases, different in very very few square inches.

Swept wings worsen the stall - the boundary layer moving toward the tips causes an ealier stall. The anhedral is a counter measure to the increased dihedral affect of the wing sweep. First it's "let's go fast", add sweep, then "too stable", so add Anhedral. Finally "now the stall is bad - no low speed", add wing fences (Soviet Aircraft), vortex generators (A-4), slats (F-86), vary the airfoil root to tip / add washout like all swept platforms (airfoil and washout need to be done together).

The landing loads are never in question - it's the high G loading that is the concern.

Dennis