Having pre landing checks completed before joining the circuit, though not technically wrong, may not be ideal either. Depending upon the type, it could mean dragging the plane into the circuit with the wheels and flaps extended, and the prop at full fine. Needlessly inefficient, and noisy for people on the ground. I generally feel more comfortable about the need for a lookout being a bit more relaxed, once I'm established in the circuit. Once I have confirmed my awareness of other traffic, their location/relative position is a bit more predictable than away from the airport environment, where altitudes and directions can be more random. That gives me a better feeling about accomplishing "head in" pre landing checks, while flying where I'm predicted to be, around other airplanes, whose position I can predict.
Ultimately, for the amphibian training I do, I would not accept a pre landing landing gear position check as being complete before entering the downwind. In my mind, by the time you get to short final, it's too stale. For landing gear position confirmation, it's pre landing for me....