The major risk in the circuit is not usually the traffic who is already there, but the traffic that is joining, that is when radio calls and standard routes come in useful. If they aren't following these and are non radio, what warning do you get then? That will obviously increase your chances of not noticing the traffic.
Whilst in the open FIR the helpfulness of the radio diminishes, so changing the viz limits wouldn't be of a huge amount of use, in that case shouldn't we have higher limits for areas of potential traffic congestion? Not very sensible really.
The scenario you mention about seeing 2 a/c is a valid one, but I was told a story today about a problem caused by an idiot instructor in a radio equipped a/c conflicting with a solo student in a non radio microlight. Was this down to a lack of awareness? yes it was and it goes down to stick&rudders point that some do place too much reliance on the radio, but I think we all agree that the radio is an important tool for traffic awareness.
Put it this way, would the London TMA operate safely on a see and avoid principle? Even on a perfectly clear day? How about getting rid of TCAS?
Standard calls and using your loaf are ideal, especially since some pilots do tend to flap if another a/c comes within 50 miles of them, a quick call to reassure them that you can see them or to say exactly where you are can help relax a fractious mind, but if the circuit is busy and there is lots of 'normal' calls, then that is neither the time or the place.