I agree. I do not think that bases functions currently allows for this.
Some time ago, there was a FR (https://forum.obsidian.md/t/embedded-base-should-have-equal-content-to-non-embedded-base) to add similar variables like this, for example the .base file itself. You would require something similar, that returns the file, another file is embedded in.
As a workaround, what about doing it like this:
![[footer]]
![[Base.base]]
Then you have the footer and the base in your notes, this should work correctly and you can do changes to the base in one place and it will affect all notes.