A "language-independent" representation of communicable concepts would be a very handy tool to have
Handy, maybe, but I believe intrinsically impossible - even mathematics is a language, with its own dialects to help obscure meaning.
Language is the medium of thought as well as of communication. To pick a language is to constrain the concepts you can consider and communicate. To pick no language is to throw out thought and communication altogether.
Note also that "language" is not just words; consider a simple image:
To a mathematician, this may be a simple Venn diagram representing a single non-universal set; to someone from Japan it might be a stylised representation of the flag, with overtones of nation, history and the Emperor. The observer cannot avoid imbuing the message with their own knowledge and their very self.
The best we can hope for, I believe, is to develop a language that is richly expressive yet easy to learn; part of my hopes for future software development is to move toward that goal by throwing off the linguistic shackles of a linear stream of words.