In my haul of presents this year there was a copy of NIL, so I embarked on reading it root-by-root. The first one is *bhag (NIL 1-2), and looking at the evidence for nominal derivations listed, I got a few ideas, which I’ll share below.
1) The root has abundant nominal derivation only in two families, Indo-Iranian and Greek. These are the same families where, according to LIV 65, verbs formed from said root are attested. Interestingly, there are no matching derivations shared by both Indo-Iranian and Greek, except the o-stem *bhago- (m.): Sanscr. bhaga- “wealth”, Iranian baga- “god, allotment“, Greek phagos “eater” (originally only found as last element of compounds).
2) Outside these families, the only attested formation is the above mentioned o-stem *bhago- (m.), found in Slavic bogъ “god” and the adjective compounds nebog- (and ubog-, not mentioned in NIL) meaning “poor”, and in Tocharian B pa:ke A pa:k “share”. Slavic also has a secondary derivation bogat- from *bagho-, formed with the productive suffix *-eH2to-. On the surface, therefore, we have three branches (Indo-Iranian, Slavic, Tocharian) showing a meaning “share, allotment, wealth”, and one branch (Greek) showing a meaning “eat”. Both NIL and LIV, following IEW and the communis opinio, take the meaning “share” to be the basic one and the Greek meaning to be a later development.
3) According to footnote 1 in LIV, the Tocharian cognates are the main reason for positing *bhag, not **bheg with a schwa secundum as the source for Greek ephagon ("ate" - suppletive aorist to esthio: "eat"). But as per footnote 8 in NIL, at least Adams in his Dictionary of Tocharian B classifies pa:ke as an Iranian loanword due to it having a plural in -nt-. Now, as, NIL states in footnote 6, it is widely assumed that Slavic bogъ loaned the meaning „god“ from Iranian. But it is also possible that the word itself with all its meanings is a loan from Iranian; after all, both meanings “god” and “wealth, allotment” are present in Iranian as well. The sound laws of Slavic don’t allow us to decide between loaned or inherited. But the fact that there are no old verbal formations based on bog- in Slavic and the absence of any cognates in Baltic, together with the identical dual semantics as in Indo-Iranian, speak, in my opinion, for bog- being a loan, not a cognate, in Slavic.
4) In that case, the Tocharian forms could not be used as evidence for the existence of /a/ as the root vowel. And instead of a three-to-one preponderance for the meaning “share”, we would have two different meanings in two different branches, as the Tocharian and Slavic correspondences to the Indo-Iranian formation, being due to loaning, not inheritance, should not be taken into account for reconstructing the original meaning.
5) If, accordingly, there is no need to reconstruct a root containing /a/, it is possible to trace both the Greek and the Indo-Iranian words back to the root *bheg “break” (LIV 66 / IEW 114-115 / NIL 6). The development “break” to “share out” in Indo-Iranian is straightforward; in the verbal system of Indo-Iranian, we would have a neat case where the meaning “break” became associated with the nasal present which, as in Baltic, was spread also to the non-present stems (at least in Vedic), while the non-nasal forms took on the meaning “share”; in Greek, the meaning changed from break” to “eat”, either via the idea of sharing food or via the idea of cutting / chewing it; in any case, the assumed development in this case is not more tortuous than the assumed development “share” > “eat”. In Greek, the family of phag- would seem to be the sole continuant of *bheg.
6) In summary, it is possible to eliminate the root *bhag “share” from the reconstruction of Indo-European, if one assumes that the Slavic and Tocharian cognates are actually loans from Iranian and that the Indo-Iranian and Greek cognates actually continue *bheg “break”.
I can’t say whether anything of the above is truly original, as I don’t have the means and the time to chase up even the references mentioned in NIL and LIV in order to see whether these thoughts have been discussed before. But as neither NIL nor LIV even mention such a possibility, I’d appreciate my readers to tell me if this has been addressed before and to point out any flaws in my reasoning.