Hi Bally Heck:
You too, ask a good question, "How do you stay visual in a jet, downwind 3 or 4 miles from an airfield?"
The answer is, I suppose, that according to the US rules, a normally recognized traffic pattern is not expected to be flown - so a "crosswind," "downwind," or "base" leg are not part of a circling approach.
As far as the US rules allowing a "lower circling minima" is concerned, I guess that would depend on "lower" than what; lower than VFR conditions, most of the times, yes; but circling minima are normally higher (in some cases significantly higher) than the associated straight-in instrument minima. Many US operators have decided, on their own, to limit circling approaches for their operation to minima equal to VFR minimums (1000 feet and 3 miles).
However, even at VFR minima, I have always wondered about the sanity of anyone wanting to deliberately take a jet (particularly the types of jets that are in air transportation service today) and wander around down amongst the "weeds!"
And ... with respect to the JAA requirements for circling approaches ... how is one to determine they ARE remaining within the circling approach area if they no longer have visual contact with the airport?