Repetitive checks with no purpose is just that.
I agree. A "check" is [hopefully] a "second look" confirmation that something appropriate to a phase of flight, has been correctly accomplished. If it's not appropriate, why do it? Prop to fine pitch? Appropriate on short final when the power is back. I agree, that in a potentially noise sensitive environment, whining along downwind does nothing beneficial. "Landing gear down"? Yes, appropriate if you're going to land a wheelplane on a suitable runway. So, that makes "Landing gear down for a runway landing" an appropriate checklist readback (which I insist on hearing out loud - or it's a go around). Because, you could be landing an RG in circumstances where having the landing gear retracted is going to work out better. Perhaps (probably) that will be a more task intensive landing, so blinding following rote of "Gear down" is likely to get you into trouble.
Nearly everything in a GA plane can wait the one or two seconds needed to pause and think about what you're doing. You have those couple of seconds to do it with forethought and logic, not just rote, then think back, or read back what you have checked. When I'm flying a plane with checklist items not required for that flight, (a recent example, a survey Caravan with many survey items on the checklist, when I'm not doing survey flying in it), I'll read the item, and respond "not required", rather than skipping it. That would also apply to flying an airplane with "landing gear" as a checklist item, when it's configured as a fixed gear plane.