ACNH: Removing a Deleted Profile's House

tl;dr To remove the house of a player whose profile is deleted from your Switch, just start up ACNH has a user who does not have a house, and link them to the deleted profile.

A while back my kids got ahold of my Switch, and created a profile (not linked to a Nintendo account, in this case) and made themselves a house (or tent) on my island in Animal Crossing: New Horizons. I deleted this profile but to my surprise, the house stayed behind. The normal process to delete a resident and their house is to launch them game with their profile, press the "minus" key on the title screen, and request to delete your data. This was no longer possible because I had deleted the anonymous profile.

The fix is actually pretty simple, if non-obvious. This took me a bit to figure out so I'm writing it here for posterity.

Claiming the house with a new profile

In short, if you have deleted user data on the island, and launch the game with a new profile (that doesn't have a house), you'll be asked if you want to link the abandoned resident with the new profile. Once you've linked them, you can use the regular process to delete the resident data from the title scree, and then delete the profile (if you created it just for this purpose).

There are a few warnings and prompts along the way. The whole process can be seen in the screenshots below.