How to update content types enheriting from the System Page Content type

Question

After importing a design package I found out that my content types was missing some columns. I could add the missing columns to my "page" and "Article Page" site content type. When adding the exisiting columns I could choose to update
everything that inherits from this type:
This is working. However I still miss 2 columns in the "Page" Content type:
1. Rollup Image
2. Hide physical URLs from search
If I roll back my dev environment I can see that these are inherited from the "System Page" content type which is not editable. The columns are in the "System Page" content type but They are gone in "Page" content type that
should inherit the columns.
So how do I update these?
Thanks

Answer 1

Hi Alex,
Thank you for replying. I totally agree with you regarding not to modify the OOB content types. The content types I have created/modified is also my own "new" content types.
However the designpackage import messed up the OOB content types somehow so I'm only trying to get back to the state before the import. The only thing I see is still missing is the two previously mentioned columns in the "page" content type.
Do you have any other suggestions in how to get back to the "OOB State"?
Thank you
Update: this is my page content type. It's not set to read only so it should update the last two missing columns from its parent (system page content type) right?