You're raising interesting and quite complex issues.
You have to keep in mind that people involved in PIE studies fall into two main categories: Indo-Europeanists and Macro-Comparatists. These two categories do not have the same needs, for that matter, they are not necessarily in agreement over what PIE is or should be.
PIE is a model, like there are models for particles in physics.
Indo-Europeanists are mostly interested in etymologizing Indo-European languages, and they are as a rule quite happy with PIE as it is for example described in Fortson's book on PIE language and culture. Macro-Comparatists need PIE to be as realistic as possible, because a bad model of PIE means that comparanda will be impossible to find. Of course, Indo-Europeanists don't really care about that kind of macro-comparative issue.
For example, taking the Glottalic theory, Indo-Europeanists have not adopted it, and it is (was) quite predictable that they would reject it, because it makes glottalic "PIE" look odd, and two centuries of literature over PIE would become half unreadable. The benefits of adopting the Glottalic theory is about zero and the drawbacks are enormous.
On the contrary, Glottalic PIE is more palatable for Macro-Comparatists, because they don't have to read two centuries of literature. Besides, there is indeed very good reasons to think that the traditional series *t *d *dh is unrealistic. Especially *d, which doubtless was not phonetically voiced, but was something else. My own choice is preglottalized voiceless. Unsurprisingly, people like Bomhard who works on Nostratic have adopted the Glottalic approach of PIE, nearly from day 1.
Another point is that the current model of "PIE" has its own historiography and it is in many ways glued into its past.
"PIE" can be periodized as follows:
v0 was Old Indian
v1 was Proto-Sanskrit improved by the vowels of Greek (1850-1930)
v2 was Proto-Sanskrit improved by the vowels of Greek and the laryngeals of Hittite (1930- to this day)
Besides, there are two v2: one with 3 laryngeals (mainstream), another with 4
So what most Indo-Europeanists use is Improved-Proto-Sanskrit (IPS) v2+L3, as their standard model for "PIE". A minority works with IPS v2+L4.
At this point, we should introduce a distinction between Definitional PIE and Comparative PIE. Definitional PIE is the last stage ancestral to all IEan languages, Comparative PIE is what Indo-Europeanists routinely use as their standard model for PIE.
There is no doubt that Def. PIE and Comp. PIE do not coincide. Fortson's book on (Comp.) PIE clearly indicates that there are problems with Comp. PIE, in particular when it comes to Anatolian languages, especially the verbal system of Hittite.
So, to summarize, Comp. PIE = IPS v2+L3 but ≠ Def. PIE.
Now, to come to more concrete issues, the series *t *d *dh is what is left after Old Indian *th is removed. People are used to that "system", but it is not exactly a system, it's more a set of comparatemes, namely a set of symbols that work quite nicely for etymological purposes. That set of comparatemes can be easily converted into actual words thanks to sound laws and competent people know a good of number of sound laws and agree on which sound laws are most probably acceptable.
Typologically, the set of comparatemes *t *d *dh makes about no sense, as a potential system of phonemes, but as far as etymology is concerned, WTF with typology if it works.
Similar problems occur with vocalic comparatemes *o *a *e. Most of the 19th century was spent on sorting out the short a and long ā of Old Indian. The set of comparatemes *o *a *e is the historical result of that process of sorting out.
But if you look at the distributional properties of each comparateme, it is glaring that they cannot be actual phonemes of a system. The situation is more that a kind of castle of (phonetic) cards accounts for sound correspondences across the family.
Besides the set of comparatemes *o *a *e is somehow a compromise between the v1 and v2. If you apply rigid phonological principles, that set would probably be reduced to only two units: unstressed *a and stressed *a (plus four laryngeals).
Indeed, if you look at Indo-European languages as they really are, most of them oppose two main units *a/o to *e. Only Greek has a full-fledged distinction between three units, Italic and Celtic have a soft spot in distinguishing *a from *o.
Again, we're back to practical issues: WTF with phonology if *o *a *e works pretty nice.
It is probable that Indo-Europeanists would consider a model of PIE with only unstressed *a and stressed *a (plus four laryngeals) as being the ancestor of their Comparative PIE with three comparatemes *o *a *e, but the truth is rather that their Comparative PIE is a mirage, a fiction. But this fiction is quite practical and people are used to it, and it's been in use for about one century now. So I doubt that people will readily give it up for a more abstract model of PIE with only unstressed *a and stressed *a, that would oblige them to rewrite all the sound laws they already know.