Lots (and I mean lots) of ink has been spent going over the possible etymology of this root – so far with no firm conclusions.
A recent ‘current state of affairs’ treatment not only of *bʰréh₂tēr, but of all the primary and secondary kinship terms in Indo-European, is Birgit Anette Olsen’s long chapter in the book Kin, clan and community in Proto-Indo-European society (pp. 39–180) which came out in 2020 (full disclosure: I am one of the co-editors of the book). She meticulously sifts through the actually attested evidence and (on p. 69ff.) weighs in on various suggested origins of the word. [Note that Olsen writes out laryngeal colouring on vowels, so *ah₂ and *oh₃ instead of *eh₂ and *eh₃.]
She describes in some detail – and rejects – three prior attempts at etymologising the root. Firstly, probably the most widespread theory (I’m not actually sure who first came up with it; Szemerényi as cited here is not the origin, he just describes it):
a widespread explanation implies an association with the agent noun Ved. bhártar- ‘bearer, preserver, protector, master, husband’ (cf. Szemerényi 1977: 24), though the formal details are quite obscure and the semantic justification not much better since we would rather think of the ‘supporter’ as either the father or the husband.
Secondly, Szemerényi’s own hypothesis:
bhrātēr is not an agent noun or kinship-term formed with the suffix ‑ter‑ but consists of two segments, *bʰr- and *āter-. There can be no doubt that *bʰr- represents the nil-grade form of *bʰer- “carry, take, bring”, but what is *āter-? There again the answer seems unequivocal: it can only be the noun *āter- [i.e. ‘fire’]…
Consequently, the “brother” allegedly denoted “a person who tended the fire, looked after it, and no doubt procured the fuel as well”, a function originally delegated to the young men of the family. Here the problem is not only the extremely speculative semantic development, but also the unique compositional type of a zero-grade deverbal root noun + direct object.
Thirdly, a theory forwarded by Georges-Jean Pinault:
According to Pinault’s ingenious analysis (2007: 276f., cf. also Pinault 2017a: 89), the starting point is also the root *bʰer‑, but this time in the meaning ‘bear, give birth’. Allegedly *bʰreh₂-tr̥- is originally an adverb with zero grade of the unaccented suffix and the meaning ‘belonging to the group of (male) children’ where *bʰreh₂- is a collective “referring to the group of males borne by the same mother”. This explanation is problematic in several respects: on the semantic level, there is no evidence to suggest that “brothers” in the Indo-European sense originally had to have the same mother, the male line being of prime importance. Moreover, it is formally disturbing that a zero-grade root noun (or thematic stem?) with corresponding collective based on the root *bʰer- is not otherwise attested except as the final member of compounds such as Gk. δίφρος ‘seat, chariot-board’ < *du̯i-bʰro- ‘two-carrier’. The function of the suffix *‑tr- is rather vague, and in general the explanation implies too many unparalleled steps to be immediately convincing.
Having discarded these as involving too many formal and semantic difficulties to be convincing, and being at that point fairly willing to chalk it up to ‘nobody knows’, she does venture a suggestion of her own, which has at least fewer formal and semantic difficulties. Quoting at some length:
Overall, it may be futile to engage in far-fetched etymological speculations concerning the basic kinship terms, but if ‘brother’ is an agent noun, it can at least only partially be patterned on ‘father’ since the accent type is different. Most likely we are dealing with an original acrostatic paradigm *bʰráh₂-tōr, *bʰráh₂-tr̥-s, like (*déh₃-tōr >) *dóh₃-tōr, *dóh₃-tr̥-s ‘a (habitual) giver’, later remodelled to *bʰráh₂-tēr after *ph₂tḗr. An agent noun would imply that the derivational basis was either a verbal root or a derived verbal stem and, not to forget, the ‘brother’ would be characterized as someone actually carrying out an action.
The next question is what sort of action would be associated with ‘brothers’. At any rate, *bʰer- ‘carry’ should probably be excluded, firstly because the notion of ‘bearing’ or ‘carrying’ would seem more natural in connection with sons than brothers, and even sons are ‘born, carried’ rather than ‘carriers’; and secondly, because a denominative based on an *‑ah₂-stem in this case would probably have o-grade in the root, cf. Gk. φορά ‘carrying, burden’.
There are, however, two roots registered in LIV² as ending in *‑rH: (1) *bʰerH- (LIV² 80) “mit scharfem Werkzeug bearbeiten”[Footnote: Cognate verbs include Lat. feriō ‘hit’ and ON berja ‘beat, hit…’] and the semantically less likely (2) *bʰerh₂- (LIV² 81) “sich schnell bewegen”. …
This opens the way for a tentative explanation: if we define the root-final laryngeal of “*bʰerH-” as *‑h₂‑, the corresponding hysterokinetic agent noun *bʰr̥h₂-tḗr would be a ‘beater, fighter’. Now, as we have seen, it is a characteristic feature of Indo-European society that boys and young men are organized in Jugend-/Männerbünde or ‘brotherhoods’, and for this reason it is natural to perceive the concept of ‘brother’ as part of a collective. Consequently, it would have been possible to create a collective based on this agent noun which, with the expected contrastive accent, would be *bʰráh₂-tōr ‘collective of fighters’ or ‘brothers-in-arms’. Finally, the suffix would have been replaced by *‑ter- whether under the influence of the hysterokinetic singular or the word for ‘father’. Thus:
*bʰérh₂- ‘strike, fight’
⇒ *bʰr̥h₂-tér- ‘a single (habitual) fighter’
⇒ *bʰráh₂-tōr ‘a group of fighters, brothers-in-arms’
⇒ *bʰráh₂-ter- ‘a member of a brotherhood, a brother’
This is still a relatively new suggestion, and I’m not aware of anyone who has commented on it yet, whether for or against, but it has the advantage of being formally quite straightforward – the only part that’s not fully regular is the reshaping to *-tēr under the influence of *ph₂tḗr – and semantically quite defensible.
Of the possible candidates we have for an etymology underlying *bʰráh₂tēr, this to me has the fewest disadvantages and fits best with what we actually know about Indo-European social and family life.