Eh I'm not sure about the weekly water changes on a nano. Too much fluctuation in params. Bi weekly water changes with skimmer attached is the way to go
I had a 24g Aquapod that was skimmerless because no decent skimmer would fit in the back compartments. It did okay for the 1st 6-8 months but then a lot of nussiance algae started appearing and then it was a weekly chore to keep on top of it. When I swapped out the tank to a RSM 130, I was able to fit a Tunze 9002 skimmer in the back compartment and have not had any algae problems at all. So I would say that a skimmer will make a difference.
Got a 28 JBJ...ran no skimmer for a while. Was doing bi-weekly water changes and now upped it to monthly.
About 6 months ago I put on a Aqua C Remora...pulls dark green sludge out.
Never really had a algae problem and still dont...so really dont know if it made a difference. I guess its just piece of mind to have it running.
I have three nano, 24, 12 AND 9 gallons, with no skimmer and no algae problem. I do a 10 percent water change every other week. the other weeks , I do a 20 percent water change. I had a skimmer and it made no difference. good luck.