The mentioned pretty simple direct proof also uses Zorn's Lemma.

The paper of Cannon/Conner/Zastrow correctly proves planar compacta are aspherical. My answer/comments below sketches an alternate, strategy/technique to achieve the same result via different methods. I expect the gaps can be filled in, but the purported alternate proof has yet to be written up carefully and published.

Yes. In fact the Hilbert cube will suffice as a target, the topological countable product of closed intervals. The difficulties of the original question manifest themselves when the image of a loop is not 1 dimensional, for example if we assume all loops under consideration have image in 3 dimensional Euclidean space.

Is there a reasonable conjecture as to a plausible answer? ( For example the question at hand reminds me of the following: Are the topological spaces which underly length spaces precisely the metrizable, connected and locally path connected spaces?)