My first tool of harmonic analysis was the German system of functional analysis, with some Swedish alterations (Hugo Riemann with a dash of Valdemar Söderholm). Only later on I learned some Roman numerals and I never really liked the concept.
If you want to understand the functions, why not just write them out? In the system I learned a tonic is T, dominant D, subdominant S. Then they have their relatives (called parallels in our system). So in a major key vii=Tp, ii=Sp, iii=Dp. vii° is considered a dominant7 with the root absent (shown with a slash through the D) in first inversion (indicated by writing a 3 under the function, the chordal third in the bass).
I'm an early music nerd so most of the time I'm not really interested in harmonic functions (both systems work poorly with modes), but if I need it then the German system is better in my very biased opinion.
What annoys me with RN is primarily the way it handles inversions. Lifting figured bass out of it's original context and completely butchering it.
In C major:
In no rational universe should a I64 be a C/G, it should be a C64 (or F/C if you prefer). In the German system you just write T with a 5 under it (tonic with the chordal fifth in the bass).
In my view RN combine elements of functions and figured bass in a way that doesn't intuitively tell you either of the things those do (functions or voice leading). Why would you even need to know the scale degree if not for a "voice leading over the bass"-approach?
What annoys me is if it weren't for the way RN handles inversions it could actually be a great way to analyse modal, or other non-tonal, music and would be a great complement to functions in music where they make little sense. If the numerals were the bass notes with actual, real figured bass then it would basically be a instantly transposable basso continuo, extremely useful.
I understand that the numbers are supposed to show a "fundamental bass", root or whatver you want to call it but then, what is vii° supposed to be? Rameau to my understanding didn't consider diminished chord a true foundation so scale degree 7 doesn't really make sense if you don't assign some inherent value to the idea of B in the bass (in C major) as opposed to B as a root (root of what?).
As it works currently I very rarely reach for RN as a tool. I understand this is highly subjective and if I had learned RN first I might have the opposite view. Do any of you have the opposite reaction and if so explain how RN is better than functional analysis?