[pclmulqdq]

That wasn't directed to you, but rather toward your teacher and the OP. The intent was that as a player, I think you should go ahead and use roman numerals or jazz theory or any other form of analysis that helps you think about things. However, if you are publishing an analysis of a piece (which you did not do), you should be thinking differently than that.

As someone with a significant background in historical performance, my books of Bach preludes are still full of Roman numerals because that is a really good way to "compress" information about what notes to play.

[AlbertCory]

OK. For sure I wouldn't publish that set of chords, since anyone could create it as a shorthand.

I'm actually interested in other ways of analyzing it.

[pclmulqdq]

I mentioned this in a sister comment, but I will add:

* Schenkerian analysis is probably the modern method of looking at Bach. A sister comment also indicates that Schenker was a controversial figure (he was a German fascist), but I believe the follow-ups from other theorists on his methods are the modern ones to use for analysis.

* Analyzing otherwise in the context of counterpoint and intervals - see Gradus ad Parnassum and Kennan's book on counterpoint.

[bonzini]

I guess you could use figured bass notation. It's more complex than modern chord notation because it tells you which inversion to use, and it notates secondary dominants (e.g. V of V, which you'd write simply as D7 in jazz notation if you're in C). Personally I know too little about it to use it effectively and therefore I would stick with modern chord notation even when talking about classical pieces.