If the ATZ is 2nm radius, can an aircraft on a 4nm final be considered to be in the circuit? An aircraft joining must comply with the existing traffic pattern.
This is where the military "deadside join" (not an overhead join) has an advantage. It was designed for a busy circuit. The joining aircraft avoids the liveside pattern, by flying parallel to the duty runway, on the deadside at circuit height and fits into the pattern on the crosswind end. If anyone has to extend, it's the joining aircraft and he goes further upwind. This avoids anyone having to fly the dreaded "extended down wind leg", which gets longer and longer as more aircraft have to follow suit.