What is the conclusion on which parts of these theories are correct?
They are both correct. They are different ways of looking at the same music.
In modern harmonic syntax studies, are we only speaking of the root note or entire chords (and inversions)? By modern harmonic syntax I’m speaking of books like Tonal Harmony by Stefan Kostka (the chord progression chart), Harmony by Walter Piston, or root progression studies by Schoenberg and Yitzhak Sadai.
Modern harmonic theory generally builds on Rameau. In C major, a chord comprising C-E-G is designated
I. But there is nonetheless a recognition that inversion is significant.
Depending on the bass, it could be
The discussions on reddit and soundonsound both contain a fair number of factual inaccuracies, some of which are quite egregious (for example, the statement that Rameau's theories were published after J. S. Bach died), as well as highly questionable statements of opinion (for example, "Bach and his son were superior in knowledge and composition to Rameau"). I wouldn't give them too much credence.
Both primary sources and secondary sources with citations seem to be sorely lacking on the internet, but I did find an interesting secondary source, a quotation from Paul Berthier, Réflexions sur la vie et l'art de Jean-Philippe Rameau (1683-1764), ed. A. et J. Picard & Cie, 1957:
Haendel est partisan résolu du Traité ; Bach, le dieu du contrepoint, se déclare contre, mais l'explique à ses élèves ; Martini cite et discute, dans son Histoire de la Musique, les principes du celebre scrittore et correspond avec lui ; Marpurg s'en inspire ; Walter le mentionne dans le Musicalisches Lexicon qui est la première en date des biographies de musiciens. En Angleterre, le Traité est traduit et étudié. Fétis pense qu'il est à l'origine du Testamen d'Euler et du Système de Tartini. En somme, on peut dire que toute l'Europe musicale l'a connu, suivi, exploité et que rien ne s'y est plus fait, dans le domaine de l'harmonie, sans passer par lui ou en se passant de lui.
Machine translation with my corrections:
Handel was a staunch supporter of the Treatise; Bach, the god of counterpoint, declares himself against it, but teaches it to his students; Martini quotes and discusses, in his History of Music, the principles of the celebre scrittore and corresponds with him; Marpurg is inspired by this; Walter mentions it in the Musicalisches Lexicon, which is the first biography of musicians. In England, the Treatise was translated and studied. Fétis believes that he is a source of Euler's _Testamen_mand Tartini's System. In short, it can be said that the whole of musical Europe has known, followed, exploited him, and that nothing has been done in the field of harmony without passing through him or without him.
The most interesting thing here is that Bach taught his students about this theory even though he disagreed with it. He can't have done that if he thought it was objectively incorrect; if this assertion is correct, he must have understood that the theory was technically sound. In that case, it is likely that he simply found that it was unhelpful to him in understanding how music works, or perhaps even distracting. The theory was published when Bach was 37 years old; his personal theoretical framework was well established. Furthermore, Bach was widely recognized as musically conservative, and this resistance to new theory is certainly consistent with that.
Consider a melody E-D-D-C and a counterpoint above, C-C-B-C. Prevailing theory describes this as a consonant sixth followed by a dissonant seventh that must resolve to a consonant sixth before moving to a consonant octave. These rules describe the music perfectly well, and such rules had done so for hundreds of years. The bass part with such a progression might be C-F-G-C or A-F-G-C. Rameau would infer, however, a "fundamental bass" of C-D-G-C or A-D-G-C. Perhaps the circle-of-fifths motion of the fundamental bass offers a unifying principle underlying the contrapuntal rules of dissonance and consonance -- obviously many people think it does, because they adopted the theory -- but that doesn't mean that the previous theory was incorrect, just as Kepler's observations about orbital motion are correct but do not give as deep an insight as do Newton's laws.
Yet, on Bach's side of the argument, harmonic thinking definitely impeded my ability to understand baroque music. I look back at some of my (literally) sophomoric attempts to write pieces in the baroque style, and they clearly suffer from this lack of understanding.
Sometimes it's easier to conceive of a stepwise passage as a series of suspensions than to shoehorn it into a particular functional harmonic progression. I find the intervallic analysis in the linked question simpler and more straightforward than the harmonic analysis, and I don't find that identifying the secondary harmonies gives any particularly meaningful insight into the composition.
To ask who's right here is like asking whether fixed or movable do is correct, whether Shakespearean or Petrarchan sonnets are correct, or whether it's correct to identify intervals by cents or frequency ratios: Both are correct. One or the other may be better in some circumstances, and reasonable people may reasonably disagree.
After posting this, I re-read the question:
I’ve read in these threads that Rameau & CPE Bach have a different opinion on wether or not more notes than the root of a chord are relevant in harmonic syntax.
This is perhaps not a great way to characterize the disagreement. The detractors of Rameau's theory argue against his technique of identifying the theoretical root by rearranging the chord as a stack of thirds. They would perhaps reject the notion of harmonic syntax altogether.
The difference lies really in how the two theories associate similar chords. Consider three progressions:
IV V I;
ii V I; and
ii(6/5) V I. Rameau would group the last two together because they have the same fundamental bass and differ only by virtue of inversion. Bach, and everyone before Rameau, would group the first and last together because they have the same actual bass (^4 ^5 ^1) and differ by virtue of which intervals above the bass are present. You might say that they disagree on what the root of the chord is more than on whether "more notes than the root are relevant."
Rameau's framework might seem perhaps to lead to a simplified harmonic framework, but it merely shifts the complexity. Before, there were more rules about how to treat various combinations of intervals above a given degree of the major or minor scale, depending on what motion might exist in the bass line. Rameau gives us the simpler "fundamental bass," but there is the additional parameter of inversion.