Several bits of additional evidence in favor of (or, if you will, additional bits of data explainable by) Aikio’s new model of the Proto-Uralic root type *a-ə seem to be found in word derivation.
1. In the appendix of his handout, a PU root *parma “gadfly” is reconstructed, regularly continued by Finnic *parma †, Mari *pårmə and Obdorsk Khanty /puurəm/. ‡ Further cognates can be included though, which show that this reconstruction appears to actually go back to a trisyllabic shape *parə-ma, and ultimately to a root *parə-, perhaps an onomatopoetic one meaning “to buzz”. These are Samic *poarō (a parallel derivative along the lines of *parə-w), and Mordvinic *purəmə. The latter is curious due to its vowel-stem structure. Normally even much heavier root types than *CVCə have been reduced to a monosyllable before a full-vowel suffix: e.g. *wajŋə-ma → *wajmə “spirit”, *kajwa-ma → *kajmə “spade”.
2. A similar case appears with the root *aðə- “to sleep”, and the derivative *aðəma “sleep”. Mansi *uuləm and Hungarian álom suggest a consonant-stem starting point *alma (*aðma?) at the Ugric level at least. Yet in Mordvinic, we again find a vowel-stem formation *udəmə, though I suppose this could be a later re-derivative from the plain verb.
I should also correct that while *čašə → *čuž “barley” seems to have undergone apocope of *-ə as expected, *śalə → *śulə “intestine” has not. All this may hint that in this stem type, there was some kind of stem vowel fortification in not only Samic but indeed in Mordvinic too.
3. A parallel to the development of “sleep” in Mansi is an etymology from the UEW that can be rehabilitated, although it is not listed by Aikio: *maŋkə “curved”. The plain root would be reflected in Samic *moaŋkē “bent object”, while Mansi *muuɣəl “around” seems to reflect something like *maŋkə-la → *maŋkla → *makla → *maɣla. It is not clear how to deal with *aśkəl(ə) → *uusəl “step” though: was this maybe adjusted to *as(kə)la at a point? ☆
4. There may be in Finnic, too, a derivative of “to sleep”, of a different type: *voodek “bed” (→ Finnish vuode, etc.) A competing Uralic etymology links this to a different root *aðʲɜ “bed” (→ Hungarian ágy, etc.) and yet another etymology proposes derivation from the Baltic loan *voota “animal hide” (→ F vuota, etc.): but since *-k is a deverbal suffix by origin (cf. e.g. lähte- “to depart” → lähde “fountain”, sito- “to bind” → side “bind”), derivation from a verb seems preferrable. Additionally, Permic *woĺ- (→ Udmurt /vaĺes/ ~ /waĺes/, Komi /voĺpaś/) with the word-initial vowel breaking *o → *wo points to original *o-a rather than *a-ə for the “bed” root (cf. *orja “slave” → Udmurt /var/ ~ /war/).
5. In Finnic we can also now set up a very straightforward derivation *karə-na → *karna “tree bark”. Cf. *karə → *koori “tree bark, crust, shell” from the plain root.
6. A potential etymology of a similar type, but perhaps more complicted, could be *šarə-wa → harva “sparse”, cf. Mordvinic *čurə “ibid.” This latter form could reflect a consonant-stem derivative *šarə-wa → *šorwa, but in light of the above parallels with “sleep” and “gadfly”, seems more likely to represent the bare root. An additional cognate is Khanty *ɬarəp “light” (with a different suffix). Furthermore: this just might be the same root as *šorwa “(to) dry” (“to become thin” → “to shrink”), ♢ found in Samic *soarvē “dead pine”, Komi /šural-/, /šurëd-/ “to dry”, and possibly Nenets /tirā-/ “to dry” (although this suggests PU *šëra- → Proto-Samoyedic *tïrå-).
† Finnish paarma and kaarna exhibit a semi-regular lengthening of open vowels before /r/ + nasal: compare also saarna “sermon”, saarni “ash tree”, vaarna “bolt”; käärme “snake”.
‡ There is also Kazym Khanty /pirəm/ which could fit together with the Finnic word as *përma; alternately though, maybe this form instead originates as a contamination of some sort with the synonymous Khanty root *peeɬəm (cognate to at least Mansi *päləm).
☆ The regular reflex of *a-ə in Mansi is clearly *aa, e.g. *aðə- → *aal- “to sleep”, *kalə- → *kaal- “to die”, *wajə → *waaj “butter”, *warə “mountain” → *waar “woods”.
♢ Apologies if I’ve absorbed this idea from someone else; I have a faint memory of seeing something along these lines somewhere before.