The obvious shortcoming is that after we leave the classical period, music and tonality becomes too complex for Roman numeral analysis to be completely useful. So, we don't need to mention 9th chords or jazz 7ths and the like, and I believe you understand that already from the question.
For classical music, Roman numeral analysis is the most widely accepted method for describing common practice harmony, and I would agree that it is useful enough that every musician should be familiar with it. Furthermore, figured bass notation is highly related to Roman numeral notation in its more basic formats, and as such is important for musicians to learn especially if they are to be performing Baroque and early music.
However, the very obvious shortcoming of Roman Numeral analysis is that it analyzes music primarily in a vertical fashion, which is contrary to how we as listeners perceive music: as changes in sound over time, i.e. horizontally.
The works of J.S. Bach have, in many cases, eluded academic consensus on how they should be analyzed for centuries, and it is in the music of J.S. Bach that other methods of analysis become truly useful.
One such method is known as the Heptachord Shift. In short, rather than analyzing the harmony and function of each chord in isolation, this method tracks changes in the heptachord (that is, the set of seven pitches that defines musical tonality, or Do through Ti) over time by classifying notes outside of the current tonality to be alterations to the existing heptachord. In doing so, it aims to more accurately approximate how we experience music by analyzing change over time. In other words, a horizontal analysis.
The resource for this system of analysis is a 2001 paper by Marianne Ploger, who developed it, titled:
Heptachord Shift: A Real-Time Approach to Tracking Tonal Modulation Employing Precepts Observed in the Works of J.S. Bach. Marianne Ploger, 2001. (link)
(Your best bet on finding the article might be the University of Michigan master's thesis archives, or by emailing the writer directly.)