The sonants in PIE have consonantal and vocalic allophones, so the consonantal sonant and the vocalic sonant are regarded as one consonant phoneme. But many daughter languages of PIE (at least at some stage) have independent consonant phonemes /j/, /v/, /r/, /l/, /m/, /n/ and vowel phonemes /i/, /ī/, /u/, /ū/ respectively. Some of them even have /r̥/, /l̥/, /m̥/, /n̥/, /r̥̄/, /l̥̄/, /m̥̄/, /n̥̄/. So when did the vocalic allophones of consonant phonemes in PIE become independent vowel phonemes? Is the disappearance of laryngeals a watershed?
Phonemes are a theoretical construct, so the answer will depend to some extent on one's theoretical preferences; note that even for PIE many scholars posit independent vowel phonemes /i u/. But basically, after PIE the analysis you describe becomes much less attractive.
The grounds for the allophonic analysis in PIE are that practically all cases of vocalic liquids and nasals appear in alternation with consonantal ones, and to a lesser extent something similar holds for the vowels [i u] and the glides [j w]; and that the alternation is patterned, so that morphophonology -- specifically, ablaut grade -- largely predicts which alternant appears where. This isn't really true for any of the daughter languages.
For the liquids and nasals, only Sanskrit preserves PIE *r̥ (and possibly marginally *l̥) as such. Otherwise the syllabic liquids and nasals changed to something else in all the attested daughter languages. So the question really boils down to the vowels/glides, and in Sanskrit also the rhotic. (Most of the daughter languages do preserve the PIE glides, but not all: Greek lost *y and in Attic also *w, so Greek is definitely out.)
So what would make an allophony analysis attractive in the daughter languages? Ideally you'd want patterned alternations of the PIE kind. The problem is that sound changes and other changes have messed up these alternations in all the languages. Their remnants are often still visible (e.g. Latin solvō - solūtus), but they're no longer predictable, because the PIE ablaut system which produced them has been eroded: synchronically you can't really talk in terms of zero grades and full grades anymore (at least not without a ton of exceptions), so the conditions for the allophony have been lost.
Sanskrit comes closest to preserving the PIE ablaut system, and Sanskrit grammarians did describe the language basically in terms of what we now call ablaut and saw a relationship between vowels and glides and between syllabic and consonantal liquids. So to some extent an allophony analysis is still arguably possible for Sanskrit (though again excluding nasals), but even there it's a lot less neat than in PIE.
To your last question, the disappearance of laryngeals comes into it indirectly in that it was one of the factors (though not the only one) in the demise of the ablaut system; and note that the long syllabic resonants, and many or maybe all instances of long *ī *ū, are now usually thought to represent a resonant/vowel+laryngeal sequence.