Looking at Indo-European studies has for a while now been giving me an impression that the usual vowel system reconstruction has unnoticed flaws in it.
They are different issues from the long-running debate on the reconstruction of the stop system, though. The traditional *i *e *a *o *u, easily attestable around the world, surely has nothing wrong in it in terms of synchronic phonology. Adding in the syllabic resonants *m̥ *n̥ *r̥ *l̥ won’t be a major typological problem, either. Rather… weird things start to pile up once we instead survey the development of this vowel system in the IE languages.
For a starting point, let’s consider Anatolian. I claim no particular expertise in the area though, so instead of getting my hands dirty with data, my commentary here follows fairly closely some short overviews by H. Craig Melchert.  He ends up positing (in an update to earlier views about a simpler 4+4 system) a vowel system almost identical to PIE for Proto-Anatolian: five short vowels *i *e *a *o *u, their long counterparts including *ā < *eh₂, as well as an unpaired long vowel *ǣ < PIE *eh₁ (early on also a later redacted “*ẹ̄” < PIE *ey). All of these yield their own distinct correspondence sets, and I would not try to claim that we need to merge or split some of these phonemes. But there are some imbalances in how some different contrasts develop. Melchert does not go into featural phonology, but if we are to trust his transcription, both *e and *o would be mid vowels. Their development tendencies however diverge. There is one general similarity: most Anatolian languages seem to show a trend of qualitatively simplifying the vowel system, towards plain *i *a *u. This is completed only in Luwian, but elsewhere, too, the mid vowels have a tendency to merge with other stuff. Short *e often yields *i in some kind of raising contexts: e.g. following *j, or when pretonic (kind of resembling Germanic). In a few other positions, there are conditional developments to *a, such as before *n in Hittite and Lydian. However, by contrast, there seems to be no evidence for a raising development *o > **u. Most Anatolian languages have generally merged *o and *ō into *a and *ā. Melchert only reports three features that allow distinguishing *o and *a:
- In Hittite, stressed *o in closed syllables yields long /ā/, while *a remains short /a/.
- In Lydian, stressed /o/ is found next to a labiovelar (either a stop *Kʷ or the glide *w).
- In Lycian, the general treatment is *o > /æ/ (transcribed e; no comment on what happens to *ō).
If, in a language family elsewhere, we were faced with two correspondence sets — one of them *a ~ *a ~ *a, the other *a/ā ~ *æ ~ *a/o — I would definitely not conclude that we are to reconstruct *a and *o respectively. And I would assume that Melchert, too, only ends up reconstructing a mid vowel *o, because this is what the second Anatolian vowel corresponds to in traditional PIE, not because the reflexes so demand. Even /o/ in Lydian looks like it might represent some kind of an assimilation from the adjacent labiovelars, rather than the preservation of original rounding.
The long vowel situation seems even more worrying. We would definitely expect to see a raising *ō > **ū at least somewhere, at minimum in languages like Lycian or Luwian where *ē > *ī, if these two had made up a similar class of long mid vowels. But apparently we only get /ā/ everywhere. Melchert reports for this contrast but a single distinguishing feature: apparently *dwō- yields /dā-/ in Hittite, versus no such loss of the glide for *dwā-. This seems to me much too iffy grounds for setting up a separate *ō.
Ignoring traditional PIE for the moment and instead reconstructing *a₁ (in place of *a) versus *a₂ (in place of “*o”), there would seem to be more promising options for phonological interpretation available. In terms of height, I’d assume that it was actually the latter that was the more open vowel *[a]. This is fairly directly suggested by the different treatment in Hittite: all other things being equal, more open vowels tend to be realized as longer. No clear evidence seems to exist for a difference in backness; *a₁ remains stable-ish (though I presume there would have been some variation on if a stands for central [a] or back [ɑ]), while *a₂ has both clearly fronted (Lycian) and backed (Lydian) reflexes. This, however, provides another reason to suspect a lower value for *a₂, given that backness contrasts tend to be more labile among lower vowels.
What this seems to leave available for *a₁, then, is some kind of a weaker vowel value still prone to lowering, like [ɐ], [ɜ] or [ə], probably both [-front] and [-round]. It seems a bit curious how this has not been retained as such anywhere,  but hardly any more so than the failure of *o to surface consistently anywhere (or any other family-wide “sweep” development, such as *s > *h in early Iranian or *a *ā *u *ū > *o *a *ъ *ɨ etc. in early Slavic).
Compare to this e.g. the develoment of English short a (Early Modern /a/) and laxed u (Early Modern /ə/): the former has split over the last few centuries into a variety of lengthened (BATH lexical set, father, “tense æ”), fronted (TRAP set) and/or backed (PALM and WATER sets) reflexes, while a new neutral short /a/ is in numerous varieties filled in from earlier †/ə/ > †/ʌ/. Similar vowel histories can be found moreover e.g. in Samic varieties (old *ā > á being more heavily split in allophones etc. versus lowered *ë > â/a remaining more neutral) or in Samoyedic (old *å *a yielding a large variety of reflexes versus *ə, often lowered, remaining more neutral).
Melchert’s most recent work also mentions the recent discovery of a “new” /o/, /ō/ for several Anatolian languages, in earlier work conflated with u, ū. The short version mainly evolves from labiovelar + syllabic resonant, the long version from *aw, *ow, both also from *u next to laryngeals (thus this /ō/ corresponds to late PIE *ū, from *uH; remaining cases of Hittite /ū/ are instead from *ew, or from stressed open-syllable lengthening of *u). These are therefore clearly distinct from traditional PIE *o, *ō. If this new *o could have been in place already in Proto-Anatolian (apparently plausible at least in non-final syllables), it’s all the more reason to not suppose also the simultaneous retention of old *o.
Given that Anatolian retains numerous archaisms, and the possibility of it being the earliest split-off of Indo-European entirely, we can also proceed to ask an important question: would Proto-Anatolian *ɜ *a or traditional PIE *a *o be the more archaic state of affairs? I would end up preferring the former: a chain shift a > ɑ > o, ə > ɜ > a is more typical than the opposite.
As soon as we’ve formed this hypothesis for a “skew triangular” (or perhaps even “square”? ) vowel system *i *e *ɜ *a *u for not only Proto-Anatolian, but also Early PIE altogether, there will be numerous immediate implications. I will not go into listing all of these just yet… But to mention one, this will nicely amount to addressing the now and then raised typological objections about the rarity (and possible absense entirely before laryngeal coloring) of traditional PIE *a. In the new system, this turns out to translate into the rarity of the more marked vowel *ɜ, while the proper cardinal open vowel *a is quite frequent indeed.
 1992, “Relative Chronology and Anatolian: The Vowel System”, in Rekonstruktion und Relative Chronologie. Akten der VIII. Fachtagung der indogermanischen Gesellschaft, ed. Robert Beekes;
1993, “Historical Phonology of Anatolian” 1993, Journal of Indo-European Studies 21/3-4;
and 2015, “Hittite Historical Phonology after 100 Years (and after 20 years)“, in . I have not yet seen his 1994 monograph Anatolian Historical Phonology, but the 2015 paper seems to summarize the main points.
 In writing at least. It should be kept in mind that epigraphic evidence does not actually constitute phonetic evidence.
 Since *e may have well been half-open [ɛ] rather than half-close [e].