OK, the theory "works" when viewed in isolation, and I didn't point that out. It "works just fine" sort of like relying on duct tape to do repairs does. From a metatheoretical standpoint, the theory is suspicious because of its unnecessary complexity. It's just one way of describing what's going on, and one that obviously comes from a time before the insights that became available with nonlinear phonological theory.
The "inherent vowel" theory (no matter what the particular vowel may be) posits a general rule inserting a specific vowel to be read with each consonant. (This is the economical version: an even more complex version would list each consonant letter with the same vowel, thereby missing an obvious generalisation.) After having inserted this vowel by rule, the theory then requires a set of rules to "delete" it, and has to specify under what circumstances this "deletion" takes place. Already for Sanskrit, this fails in the face of the principle of economy (aka Occam's razor).
Compare this to a theory in which: (1) the only inherent and invariant content of any consonant letter is the consonant phoneme itself, (2) the virama simply specifies that there is no syllable rhyme (in practice, a vowel position) following a consonant, (3) vowel matras merely add their vowel to the consonant (in syllable rhyme position), and (4) if nothing else is specified, regular phonological principles of syllabification (and not special orthographic rules) determine where the phonological (not orthographic) default vowel will or will not be inserted.
For Hindi, you have to posit extra rules that count syllables to decide where the "inherent vowel" is "deleted" (after being inserted by the general "insert 'inherent' vowel" rule) rather than simply let the native reader's knowledge of phonology and syllable structure do the work. (See Lloyd Anderson's review of The World's Writing Systems for a critique of the "inherent vowel" theory and its application to Sanskrit and Hindi.) And this is just for the simple cases. For other cases involving affixation, where syllabification principles and last-resort default vowel insertion work just fine, you have to posit workarounds to get the ever creakier "inherent vowel" edifice to work. This is what was done in Kachru, Kachru and Sridhar's (2008) Language in South Asia, where they admit that their syllable counting analysis doesn't work beyond the simplest cases and refer the reader to Ohala's 1983 analysis of Hindi phonology, critiqued by Norval Smith in 1984 for its unsatisfactory nature:
"The account of ə-deletion is perhaps the least satisfactory part of the book and points up the need for a non-linear analysis, using some form of sonority hierarchy, to predict the consonant cluster environments in which ə may and may not be deleted.
The overall impression received on reading this monograph is one of irritated disappointment tempered by considerable admiration (...)".
The theory just adds unnecessary complexity that doesn't stand in the face of our current understanding of a native speaker-reader's phonological knowledge.
Without going into details here, it gets even more complicated for Batak and Lampung in Sumatra, which have a special spelling rule that shifts vowel marks from the syllable onset consonant letter onto the consonant letter that is followed by the virama.
In any case, all I wanted to know was if anyone had any idea where the outdated and unsatisfactory "inherent vowel" theory came from originally. I've managed to trace mentions of the idea to the mid-to early 1800s but the farther back you go, the vaguer people get about where their ideas come from, our modern citation conventions not yet being in vogue. On the other hand, I have a fairly good idea of where the latest illustration of this theory comes from (Ohala 1983), and where the critiques and alternative viewpoints are to be found (Smith and Anderson). I just want to get as accurate as possible in my citations while adding a more complete and balanced discussion of abugidas to the Wikipedia article(s). (As well as for a possible future journal paper on the role of phonological syllabification in script typology.)