¼λ.OK, so you said that we can look at the transmission line as loop antenna, but in that shape it will be badly matched to free space. So I would like to re-formulate the question: What decides if the antenna will match transmission line to free space? Obviously the dipole antenna is a good match. Transmission line is not.
As I said before, is it because the EM wave from one wire in transmission line is inducing the current in other wire and being almost completely "absorbed" in the wire?
There are many good antenna designs that don't use quarter wavelength conductor segments, but most of the easiest to understand do.
A half λ section gives lots of umph. A quarter λ driven conductor plus the image current in the ground plane often works out to be ½λ.
A full λ section tends to reabsorb it's own photons. ½ emits and the other ½ is 180º out of phase, so it absorbs.
There are lots of exceptions to this rule of thumb. Plus there are lots of caveats like remembering the physical wavelength varies with the dielectric or remembering gaps often count as much as conductors (slot antennas, etc.).
Yet ¼λ is a good rule of thumb.
Another rule of thumb: parts less than 1/10 λ can be ignored. Again, not always true, but it usually is. Thus your gap between wires of less than 1/10 λ makes a bad antenna. If the gap were bigger (like ½λ) that wouldn't be true. Then you would need to run the field equations to see what happens.