No, the strain gauge is deformed by the wing deforming. You can apply the load anywhere. (In fact if you tried to put a heavy weight directly on top of a strain gauge you would probably damage the gauge).
Maybe you are getting the wrong idea thinking of it as a "wire". The piece of the gauge that changes length is small, typically from a few mm long down to less than 1mm. The thickness of the "wire" is of the order of 0.01 to 0.1mm.
Of course the gauge has to to be connected to the measuring equipment by wires, and those wires are often attached to the surface of the component as well, to stop them flapping about or getting tangled up and damaged. If you look casually at a straingauged component, the connecting wires are usually more obvious visually than the gauge at the end of them, which is just a bit of thin film material covered with the glue that fixes it to the component.
If you haven't seen a gauge (but only pictures of them) they look something like the RF security tags used in library books or on expensive items in shops, but very much smaller.
If you were testing something as complex as a plane wing you would have literally hundreds of gauges attached to the wing at different places, to get a picture of the strain pattern all over the wing, but you might only apply the load at one point (for example a force at the wing tip).
For example if you wanted to measure the behaviour of a cantilever I-beam bending, you would put several gauges along the top and bottom flanges to measure the axial strain (if you think about a shear force and bending moment diagram, the stress and strain varies along the length of the beam) and possibly some more gauges on the web of the beam to measure the shear strain in it. There might be 10 or 20 gauges all together. You would then apply a single load at the tip of the beam.