Do you think the core human 'organs' for speech/sound production can be reduced to the lips, tongue and throat? I realize there are elements like nasals and glottal stops but the other three seem to be the central ones. After all, you can still get something like the 'na' sound with the nose pinched and closed but not if the tongue doesn't touch the roof of the mouth.
In a word, no. There are more organs involved in speech production than the ones you list.
Bear in mind that "the throat" is not a single organ, but a set of organs and cavities. The ones involved in speech production include the true vocal folds, which are part of the larynx, which lies below the pharynx. The true folds phonate when they are brought close together and a stream of air from the lungs flows through them to produce all voicing. The vocal folds also produce glottal consonants, and the pharynx makes the pharyngeal consonants possible. The larynx and the pharynx together also form part of the resonating chamber through which speech sounds propagate. The whole chamber is the vocal tract, which includes every space from the true vocal folds to the lips and nostrils. The shape of the vocal tract can be altered in many ways to produce many sounds thanks, not just to the tongue and velum, but to I forget how many muscles in or for its walls and organs. Yes, the cheeks are involved in speech production too.
What is more, our perception of speech sounds as sequences of discrete consecutive sounds grouped into discrete words is an illusion. It is the result of the brain's ability to decode actual speech, which is a stream of often overlapping sets of acoustic cues with no spaces between them other than the spaces created by blocking airflow entirely, and by stopping to inhale. Many of the motions for speech sounds, including the onset of voicing, whose timing makes the difference between our perception of voiced and voiceless consonants, occur at speeds most conveniently measured in microseconds.
BTW, the velum, aka "the soft palate," from which the uvula hangs, has muscles and is every bit as important to articulation as the lips, tongue, larynx, and pharynx. It can not only help to produce velar sounds like [k], [g], [x], and others; it also closes (most of the way) the aperture between the oral and nasal cavities, differentiating oral vs. nasal sounds.
We don't notice the velum because it's farther back in the mouth than the tongue and lips, and kinesthesia gets weaker the farther toward the core of the body you go. (That's why you can feel where your fingers are, but not where your pancreas is.) And, as you probably know, the uvula itself can be used to produce consonants.
In short, modeling speech production is not a simple task. (Consider, for a moment, all the neurology that I haven't mentioned.)
I suggest that you read up on speech science for more information. You can always start here at Wikipedia. https://en.wikipedia.org/wiki/Speech_science
You may be able to approximate a nasal sound with closed nostrils, but if you pinch your nose shut and try to pronounce a nasal sound you will find you cannot do it. You can feel the air building up behind where you have pinched it shut.
Also, the nasal cavity acts as a secondary resonator. It is involved in producing nasal vowels from e.g. French and Portuguese.
Speech synthesis works on the acoustic signal. The formant synthesis approach is based on Linear Predictive Coding (LPC) techniques from speech signal compression studies. These use a source-filter model.
You can divide the signal into voiced (where the vocal chords are vibrating, producing an impulse train) and unvoiced (where the source is a chaotic airflow, usually modelled by a random number generator). This forms the source. Some speech synthesizer voices use a signal derived from the original audio recording as the source -- for example in Residue-Excited LPC in the diphone voices from festival.
In LPC analysis, LPC coefficients are produced that work to reconstruct the acoustic signal. From these coefficients, the formants (peaks) of the acoustic signal can be derived. Using either the LPC coefficients or the formants produces the same effect -- to filter the source signal, reconstructing the original.
The first two formants (F1 and F2) indicate the vowel on the IPA vowel chart being produced, and are used this way in acoustic speech analysis. The other formants add richness and additional harmonics to the phoneme.
There are several techniques for producing the speech signal using the LPC/formant data:
Use the LPC coefficients and residue (remaining signal) -- this is used in concatenative synthesizers like Festival and Flite.
Use the formants, storing them for each segment of sound -- this is used by e.g. the Klatt speech synthesizers.
Derive the formants from target F1 & F2 formants for a given vowel -- this is used by e.g. the rsynth speech synthesizer.
Model the vocal tract as a cylindrical tube, with the width of each section derived from the F1 & F2 formants -- this is used by e.g. Gnuspeech.
Use Hidden Markov Models (HMM) to predict the speech parameters (pitch, duration, etc.) including either the LPC coefficients or the derived formant data -- this is used by the HTS and clustergen based voices from Festival and Flite.
The source is modelling the airflow from the lungs and the vocal chords. The filter is modelling the effect the shape of the path the air travels through the mouth and nasal cavity, controlled by things like tongue shape and jaw height.
This does not cover the effect of increasing the force of the air through the lungs (volume), closing the mouth (for stops), aspiration and other effects or for non-pulmonic consonants.