This is a very short read, but very condensed and well-written. There's two topics examined closely, one is that Beakers spread Old European (or the Alteuropaisch) of Krahe's hydronymy. The other is the possibility that some Beaker communities spoke something ancestral to Celtic, being more or less a proxy.
Beakers have been variously implicated in the origin of the old European hydronymy because after all, what other archaeological horizon has unified such a vast region? But there are problems with this possibility as Falileyev lays out, the least of which is the old naming system is theoretically larger that the effective area Beakers are thought to have inhabited, assuming 1) any unity existed among Beakers 2) or that they were sufficiently present for enough time to change an existing hydronymy.
I think Falileyev really channels us into one conclusion: the sporadically settled Beaker phenomenon has no plausible role in creating the European hydronymy. In a paradoxical sense, the stability of the hydronyms over the last 2,500 years really questions their vulnerability to begin with, so it's hard to seriously accept the possibility that an ethnic minority changed all naming conventions, then after crystallizing, those names remaining stable through all the turmoil over the last 4,000 years.
I'll also throw this out there, that Beakers post-date putative IE speakers in much of the zone in which the area this hydronymy exists. Really, the smart money is implicating the earliest Near Eastern farmers of Europe.
First, let's accept that the Celtic question really is not about the Celtic family in the regular sense. Here we are using it as a proxy for a language that no longer exists. Celtic itself, as Falileyev explains, is also a developing story and it may have no relation whatsoever with the West and certainly not the Beaker people. Or it could.
The Beakerfolk don't exactly help with this question either. If Bell Beaker had drove in from the Pontic Steppe and left a neat trail of breadcrumbs across Eastern Europe, then maybe this question would be less difficult. Instead Beakers come from the other direction, at least in their immediate history. It also doesn't help that many archaeologists reject questions of ethnicity, especially as it relates to cultural and linguistic transmission.
From a glottochronogical standpoint, in many schemes of mainstream linguistics IE-speaking Beakers is still doable. Celtic and/or Italic are sufficiently old enough to have been seeded by early Martime Beakers and rising to the top of the primordial foam by the Urnfield period. Many of the phylogenetic trees place them and Tocharian at the base, or the most diverged after Anatolian.
Aside from the favorable clockwork, it's also noteworthy that Tocharian and I-C are physically the most distant of the recorded European languages. In this sense, although the Beaker story is strange, it offers a framework that other archaeological periods fail to demonstrate.
Falileyev criticizes the 'Celtic from the West' as argued by Koch and Cunliffe, seeing a number of weak points that can't be reconciled. He does however give a preview of a work by Gibson and Wodtko that he thinks is more promising in which a linguistic mosaic was formed by pockets of Beaker settlements over the continent and this theoretical language possibly becoming ancestral to Celtic (or not exluding Celtic). In this scenario there is no nucleus or geographic origin of Celtic, or whatever, just pockets of communities that speak a similar language.
I'll look for the Gibson and Wodtko work. It looks interesting [link].