thank you very much!

yes, I definitely agree with you that it's the TT'ing day by day that uncements the villagers eventually. but since changing your "boxes day" means you
have to begin TT'ing day by day, it's essentially the same thing; I just thought it might be simpler explained that way.
I'm not entirely sure what you mean on the second part, though. my cycling towns are always at 9 villagers except, obviously, after one has been adopted -- then it drops back to eight when you TT back a year and five days, a ninth plots when you TT ahead a year, and then on your third load of the game when you're back to your "boxes day", you're back at 9 villagers with one moving out. so I'm thinking you meant don't let your town get to 10 villagers...? but I'm still not 100% sure what you mean by this being a bad thing. I don't keep track of my villager count when I'm cycling, (in fact I usually don't do anything but save & quit immediately) but i never encounter any problems even when I pick up a void. it simply overwrites the random move-in that plots on your second load of the game during cycling. in fact, for that reason I'm pretty sure you won't get to ten villagers at all using the cycling method I use unless you were to go to someone's town and invite a tenth over. but even then, I'm still pretty sure it evens itself out, as it does when you invite a camper.
I'm thinking that since you use a different cycling method, maybe you encounter issues when I don't? i really can't say for sure, but I just know I've never had any problems with my villager count using this method, even if I pick up voids or invite campers.