The two things I like about the TLR are: square format and waistlevel composing.
I'm a sucker for square format, and like to compose on a square viewscreen, as opposed to cropping a rectangular frame later. To me, the square is the photographic version of the perfect circle; nether preferentially biasing the image toward the horizontal or the vertical, and is ideal for abstraction.
As others have stated, the waistlevel composing lends an air of anonymity to one's street photography that is hard to experience with any other kind of camera. With a whisper-quiet leaf shutter up front, one can make an exposure undetected whilst standing virtually next to the subject.
The one thing that TLR cameras don't give you is the ability to anticipate the subject's movement, as one would in the area outside of the framelines of a rangefinder, so in this respect they remain reflex cameras at the core. Yet rangefinders themselves share another trait with SLR's that TLR's don't: having to put the camera up to your face to make an exposure.
As for HCB, his use of the Leica as an 'extension of his eye' wouldn't work with a TLR, since you have to glance down toward your shoes to see the image on the ground glass of a TLR, away from the subject entirely. It simply didn't fit into his working methodology, which was to see the composition coming together, prior to putting the camera up to his eye.
~Joe