I'm not a linguist but it seems that most phonemes exist as areas (or multiple discontinuous areas as seen in TFA) in phonetic space while IPA attempts to describe only points in phonetic space[0]. Wouldn't it be better to instead just define X = {(F1min,F2min,...),(F1max,F2max,...)} for every dialect instead of making a universal X = (F1,F2,...) and then using it describe any area in phonetic space that happens to contain it?
IPA seems to describe areas as far as I can tell; each consonant is how-do-you-make-the-sound and where-is-the-sound-made, not specific frequencies, charted on a grid. The vowel chart is "where is your tongue in your mouth".
But it seems that the "where the sound is made" aspect is being described as discrete whereas the positioning approaches more of a spectrum. Even if it were discrete, the fact that most phonemes can be produced with multiple positions would require at least a list of points.
IPA is about phones, not phonemes. Phones are described precisely and are (mostly [1]) constant across all languages; phonemes are broader and are language-specific.
Multiple phones can belong to a single phoneme within a given language. For example, English groups the aspirated stops with their non-aspirated counterparts (e.g., [pʰ] and [p] both belong to /p/ because they are non-contrastive, despite being different sounds phonetically).
This is why it is the International Phonetic Alphabet, and not the International Phonemic Alphabet.
[1] Worth noting that there can be some variance among different speakers when it comes to the articulation of specific phones, but the IPA essentially is the result of determining whether languages draw any meaningful distinctions among these. If there are separate symbols in IPA, the phones are noticeably distinct. It is very uncommon for multiple sounds to get mapped to a single phone in IPA by rule, and this usually happens based on some dispute among linguists. A good example might be the "tensed" consonants of Korean, which have their own diacritic applied but are not fully understood (meaning linguists cannot precisely identify what, if anything, separates them from their un-tensed counterparts).
Phones are pretty consistent, but I think the biggest piece of the puzzle is that different languages don't care about all the differences. A phone exists in IPA if a language cares about the difference.
For example, if you pronounce "sju sjösjuka sjömän" with [ʃ] instead of [ɧ], I'll understand you just fine, but I'll instantly know that you aren't a native Swedish speaker. (Or, you are, and you're speaking a Swedish dialect that has replaced [ɧ] with [ʃ], because of course that's also a thing)
But if you mis-pronounce "skön" (nice) as [ʃøːn] instead of [ɧøːn], you're getting awfully close to [ɕøːn], which is how you pronounce "kön" in Swedish, which means gender. So as a Swedish speaker, the phoneme has changed, I no longer know if you're saying "skön" or "kön", or maybe "schön" in German? Swedish "cares" about this difference. But you, as an English speaker, might not be able to hear the difference.
And if there was a hypothetical language that had even further subdivisions of these sounds, speakers of that language would be upset that neither you nor me could tell the difference between two sounds that speakers of that language care about.
So the mapping of phones to phonemes is highly language (and dialect) dependent, and you should think about it in terms of continuous ranges or tolerances, instead of a discrete 1:n mapping.
[0] https://sail.usc.edu/~lgoldste/General_Phonetics/Source_Filt...