I know that many people like to, or are required to, use written checklists but I think the average PPL in a bug smasher should be more than capable of committing key checklists to memory, minimizing the amount of time that eyes are inside the cockpit within the circuit.
Having hired aircraft at many places I've lived around the world, I found very few instructors/check pilots took issue with the fairly standard lists I used for common types. My impression was that they were much more concerned to see that important checks were being done, rather than nit-picking specifics. Of course, there were some location-specifc items that an Oz flatlander like me had to add in e.g. Alps flying.
Nowadays with my own aircraft, I do have written checklists, POH etc as required, but I still exercise my brain with the memorized list, making a point of e.g. re-checking just 4 items before entering the runway. I appreciate that this is not the way to do it for all operations, including most commercial flying. But to my mind, nothing looks worse than having a PPL fumbling around with a written checklist on downwind.
To do a bit of nit-picking myself, I don't like this "down and locked" business. How do you know that? All you've got in the cockpit in most light aircraft are "3 greens" (or whatever) and that's what you're checking, as a proxy for all being well with your wheels.