Hi,
It is very strange and I have never encountered this before - nothing happens just so, there must be some pattern.
I do want the same web part visible on another page and wondered if that was the problem but I have just done some testing and it seems that I can have the same static HTML web part visible on some pages and not others. (I even cleared the cache on the server and browser to make sure.)
Do the pages have the same page template? If so, the web part setting is shared across page template.
Anyway, from your description everything seems to be OK. But in order to investigate this issue, would it be possible to get the backups of the project folder and database together with the export package so we can follow your steps, reproduce and investigate the issue (you can send the backups to support@kentico.com, or links to download them).
Thank you.
Best regards,
Juraj Ondrus