First off, yes; it is easily possible to add a table into another table that’s inside a module. You simply need to understand what table.insert() does and how to insert something into a table without needing table.insert() to do so.
I deleted this post entire content because the question was mainly based off my own ignorance on tables and how they work. It was answered in my other post; How do you insert a full table into another table/How do you change the name of a table?.
Q: Is it possible to add a table into another table that’s inside a module?
A: Yes.
As for why your script isn’t working, it’s hard to tell upon first glance. Is the Inserted warning firing and the tables printing?
If so, then I would verify you are inserting into module tables directly and not inserting into table references from a different module. (I don’t know if you can pass a module table from a different module and still have the changes apply to the master module since the table variable might end up just being a reference and editing a reference isn’t going to make changes to the source, if that makes sense.)
Edit: I tested this functionality and it doesn’t matter if a table reference within a different modulescript is edited, it still applies the changes to the original source. Referencing between multiple modules carries the original table, unlike how I originally thought.