The issue occurs with fully generated villages too in my testing. The screenshot is due to partial generation, if you regenerate a village that was generated pre 1.16 in a 1.16 world it will look different.
Edit: Never mind, this doesn't really add anything, now that I understand the jigsaw block it makes sense that it would be having this issue.
Looks like this may also affect villages in general in upgrades from 1.15.2 -> 1.16.2. Unknown the specific circumstances as to when though.
I've found villages which seem to generate the issue, deleting the chunks associated which don't seem to have pillager outposts seems to resolve it. The specific village was in the savanna I think.
It was confirmed on 25th of June (over a month ago) to also affect vanilla Minecraft. It's literally the 2nd or 3rd response.
So lately I've been experimenting with upgrading my server for my users. This bug seems to only occur when chunks that contain villages are loaded that were generated prior to the 1.16.1 upgrade. Since I run a modded server, I have used the WorldEdit beta to delete the chunks those villages are in and a reasonable area around them to account for changes to the generator.
This seems to remove the messages. Your mileage may vary, and you have to consider that doing this may remove content created by your users. How to do this on vanilla would be deleting the chunks in an editor while offline and reloading it.. though there may be other methods.
World generated on 1.15.2, upgraded to 1.16.2/when upgraded to 1.16.2 that has the issue: https://drive.google.com/file/d/130h8UJWViF-YL_UKjszo1RJlsJdabzMc/view?usp=sharing
Seed: 1234
Coords: -415 70 114