The tags include "phonetics" but not "phonology", but people often ask questions about phonetics and tag them "phonology", so I assume that you're asking about a phonological issue. If you really mean specifically and only "as a question of phonetics, to be answered in strictly phonetic terms", I refer you to myriad statements by the phonetician John Ohala which can be summed up by asking "Where do you see syllables in the phonetic facts?". Syllabification is not a phonetic matter that can be transcribed in the same way that aspiration (VOT), length (duration) etc can be. It is purely phonological (if we assume the syllable, and such-and-such structure, we can efficiently account for the observable facts of speech – we should also inquire about the analysis that omits the syllable, since it could be redundant).
There are two questions here, one about "arguments against", and one about "why not universally accepted". The answer to the latter is pretty straightforward. Universal acceptance is an unattainable goal: even wide acceptance is close to non-existent. The reasons for non-acceptance are going to be myriad, and will especially be sensitive to one's philosophical "primaries" in linguistic research. Some linguists take descriptive adequacy to be the most important consideration and abjure theoretical speculation; others take theoretical enlighteningness to be primary, and are willing to dispense with a few inconvenient-looking empirical challenges (known as "problems for future research"). If Wells' account fails to cover certain facts, that would be a particular problem for people of the first type: if it fails to be stated elegantly in a well-supported linguistic theory, that would be a problem for people of the second type. (Alternatively, fail to state the analysis in such a way that it could be easily translate into such a theory). It's very hard to satisfy people who want the best of those two worlds.
It is not clear to me that I can empirically test the paper, because it depends on non-obvious factual claims. His analysis relies on a 5-point continuous stress scale, which gives meaning to "more strongly" in his basic rule of syllabification. I take "main stressed", "secondarily stressed" and "unstressed" to be fairly uncontroversial observational primaries, and question the two additional degrees. I see no way to dispose of at least three distinctions, since in a word with two stresses (ˈtarpən, ˈtæmˌpɔn, ˌlæmˈpun), you can't predict which syllable has the primary stress. However, I do not actually know if that would be his analysis of these words – we need a guide that enables us to know where stress marks are to be introduced, vs. omitted.
His distinction between 4 and 5 stress hinges on the full versus reduced vowel distinction, exemplified with substitution-product /ˌsʌb.stɪ.ˈtjuːʃ.ən.₀prɒd.ʌkt/ and magnitude /ˈmæg.nɪ.tjuːd/. The former is claimed to have the stress profile 2, 5, 1, 5, 3, 4 with [ən] having 5 strength, and the latter is X, 5, 4 (he doesn't say – I assume it is 1). The 4/5 distinction has to do with having a reduced vowel or not – how do we determine that a vowel is reduced? I assume that [ə] is always a reduced vowel in English, and if you have a schwa-like vowel that isn't reduced, you transcribe it as [ʌ]. In "magnitude", following his transcriptions, the 5-strength vowel is [ɪ], not [ə]: but not every [ɪ] is a reduced vowel (at least I don't think that is the case). I can make all sorts of conjectures about what his analysis would be for various words, but the fact is, since I do not know when he will say that a given vowel is reduced vs. unreduced, I can't empirically test his theory against a wider corpus. I simply do not know if he gets the facts right. I am inclined to assume that the system is descriptively adequate, but faith-based theory-testing is not proper science. (This could be a framework-related communication problem: he knows what he means, and so do others working in his framework, but I don't).
I presume that your underlying question is about empirically adequacy and you're not concerned with abstract theory comparison, but in case that is of interest, I will simply say that I resist the idea that theoretical entities can be created and exploited ad libitum: this is the problem with his stress-strength scale. In a similar vein, I find the condition "Phonotactic constraints on syllable structure (as established with reference to monosyllables) are not violated" to be theoretically incomprehensible, and even descriptively pretty imprecise. It is not clear to me how he gets the distinction between "militaristic" and "capitalistic", with an aspiration contrast between itʰarist and italist. We would have to assume syllabification differences i.tarist and it.alist: but how does that come about? For instance is "it" in "capitalistic" more strongly stressed, compared to "militaristic"? Or does he have some kind of "surface analogy" rule where the pronunciation of these words is determined by referring to the pronunciation of "military, capital"? No doubt he has answers to these questions, but if I don't know the answer, I can't evaluate his theory.
Overall, something like his theory looks like it is correct, since it restates the Selkirk-Ito-Mester (and others) perspective on English allophony and syllabification, indeed continues a line of analysis that Kahn could have more rigorously pursued in his analysis (the main point of difference being the treatment of "city": [sɪ.ti] for Kahn).