Villager Cycling
This method 100% guarantees the "moving-out" thought bubble from your villagers every time you do it.
(TLDR, a random villager will want to move out 20 days after a new villager moves in.)

Before You Start...
This method requires time travel. Your island WILL get weeds and you WILL lose streaks.
You may get items in your mailbox during this. Be mindful if you're against "cheating."
I have ONLY tested this on the day a villager moved IN to my island.

Step 1: Original Move-in
On the day that a villager moves in (Isabelle will announce it), go meet them in their house. They should be sweeping up and surrounded by boxes. Talk to them and then SAVE & CLOSE.

NOTE DOWN WHAT DAY IT IS. You NEED this for tracking purposes!

Step 2: Time Travel
Make sure ACNH is closed completely and then change the system time <20 days> in the future from your original move-in date.
Afterwards, launch ACNH and proceed to Step 3.

Step 3: Thought Bubble
Search your island for the villager with the thought bubble.
(You may have to enter/exit buildings or time travel an hour or 2 in the future.)
If you want to KEEP, proceed to Step 4.
If you want to EVICT, proceed to Step 5.

Step 4: Keep & Try Again
Tell them you want them to stay and then SAVE & CLOSE completely.
Go 20 days back (to original starting day), load in, save & close.
Repeat Step 2.

Step 5: Eviction
Tell them to leave and then SAVE & CLOSE completely.
Continue the move out process by taking it 1 DAY AT A TIME, saving & closing every day:
1st day: Boxes
2nd day: Vacant
3rd day: Sold (if left vacant)

Don't skip ahead, your game may bug!!!

Box Day
When your villager is "in boxes," it means that everything in their house is packed & they will be moving a day later. Isabelle should announce this.
You can either use this time to trade your villager, give them to a friend, or you can save & close to Vacant Day.

Vacant Day
The previous villager's house should be gone completely & there should be an open land plot available.
Use this time to recruit someone from a friend's island, an island tour, or whatever. You could also leave it to chance and save & close and go to Sold Day.

Sold Day
If you don't find a villager to move into the vacant plot in 24 hours, it will automatically be sold to a random villager.
If you recruited a villager during Vacant Day, Sold Day becomes their Move-In Day.

Then rinse and repeat all the steps!!!

I have tried this method 5 times now and it has worked EVERY. SINGLE. TIME!

YES, it is a little bit of a pain in the ass because you don't get to choose who wants to move, but it's the only method that I've found so far that is reminiscent of ACNL villager cycling!!!

I'm sure there is a way to polish and perfect this method, but I'm impatient and just want to get stupid rizZO OUT OF MY ISLAND so enjoy~