Does our alphabet encompass almost all possible sounds?

The question details ask for a meticulous and specific answer (though the question itself is neither).

The original 24 letter alphabet used for Latin did not even encompass the sounds of its daughter languages, let alone the sounds of other languages. Centuries of often messy digraph and diacritic solutions ensued.

But any language using a Latin alphabet as its script or as a scholarly transliteration of its script has, by definition, come up with a workable means of representing its phonemic inventory using Latin letters and diacritics.

And any phonemic alphabet or Abjad or syllabary that has needed to represent new sounds has found ways of doing so. That includes extensions of Cyrillic, Arabic, Greek, and Hebrew script.

A key restriction for those script is that they are normally only called on to represent the phonemic inventory of a language, and not the more detailed distinctions of its phonetic inventory.

If the IPA counts as an extension of the Latin alphabet, then most phonetic variation is provided for as well, and the remainder can be stabbed at with diacritics. This does not deal with the gradiation of all possible sounds that can come out of a mouth, because the IPA is not a spectrogram. But it does deal with the variation in sound that can be usefully perceived by a linguist.

The IPA in turn is only concerned with sounds that can occur in non-pathological speech. But if the IPA counts as an extension to the Latin alphabet, so do the Extensions to the International Phonetic Alphabet for disordered speech.