[The Black Rabbit]

I'd hazard a guess at a SPOTTED FLYCATCHER. Quite like a small plain robin, not rare, and ofen as you say, seem to hover around bushes/fences etc... picking up insects.
Very often the Spotted Flycatcher will take up a perch - chase a fly, return to THE SAME SPOT, chase another fly etc... etc...
If your 'dull brown small fawn robin' was doing that - I'd put a lot of money on the fact your bird was indeed a Spotted Flycatcher.