But aren't the Celtic languages Indo-European? If so, then they (the Celtic languages) would've presumably come from at least some sort of transaction with migrants from the east.
Yes. So there are two hypotheses currently as to the origins of the Indo-Europeans. One is the Kurgan hypothesis, which claims that Indo-European languages and culture were spread by the Kurgan culture which originated north of the Black Sea. This idea has been bolstered by current genetic analysis as Yamnaya DNA is highly diffused in modern Europeans. The second is the Anatolian hypothesis, which claims that Indo-Europeans originated in Anatolia. Currently, there isn't a great deal of genetic evidence for this view, but Cunliffe doesn't outright deny it as a possibility and argues that both hypotheses may be correct in that two separate waves of Indo-Europeans - one from the steppes and the other from Anatolia - may have spread Indo-European culture and languages to Europe. In either case, the Indo-Europeans unequivocally came from the east.
But we are speaking here of Proto-Indo-Europeans, not Proto-Celts. Proto-Celtic languages developed from Proto-Indo-European and did not exist at the time when the Yamnaya were charging through Europe on their horses and chariots. The genesis of Proto-Celtic, according to Koch and Cunliffe, occurred later and specifically in the Atlantic-facade of Europe. To give you an analogy: it is correct to say that Latin (and, as an aside, there are some interesting ideas about an Italo-Celtic ancestral language from which both Proto-Italic and Proto-Celtic may descend) is an Indo-European language, but it would not be right to say that Latin "came from the east" or that the Latins themselves settled the Italic Peninsula, since the genesis of the Latin language and the ethnogenesis of the Latins themselves occurred in situ in Italy. In other words, Latin is native to Italy, even if it is the product of an earlier culture and language that certainly did not originate in Italy.
In 2015 there was a DNA comparison made between a Neolithic corpse in Ireland, Bronze Age corpses from Ireland, and modern Irish people—and the Bronze Age and the modern Irish had the greater affinity. Wouldn't that evince a Bronze Age migration into Ireland, presumably bringing with it Indo-European language?
This is a really, really complicated issue. So everyone in Europe today descends from several different prehistoric populations: (1) Western Hunter Gatherers, or WHGs, who are the closest thing to "indigenous" Europeans; (2) Neolithic Farmers, sometimes called Early European Farmers, or EEF/ENF, who came to Europe probably from Anatolia and, as you can guess, brought with them agriculture; and (3) the Yamnaya, or Indo-Europeans. Every population in Europe is the product of these three groups, to varying degrees:

Now what's interesting about this graph is the lower portion, which shows the degree of genetic legacy each of these groups had on ancient peoples of sort that was found in Ireland. The Neolithic woman referenced in the study obviously belonged to the Neolithic Farmers group, and her DNA analysis would probably look something like that of the Starčevo body in the above graph. Starčevo was a very old culture that predated the Atlantic Bronze Age by ~5,000 years. The Rathlin man mentioned in the study was probably descended from all three groups, and thus would cluster genetically closer to modern Irish populations who are also the product of all three groups. The reason why the Neolithic woman would have clustered closer to modern Spaniards is because the ENFs had a far greater genetic impact in Spain (and in all of southern Europe) than they did in Ireland.
All of this is to say that the Irish, including the Irish language, have several groups of ancestors, many of whom migrated to Ireland from the east. However, Irish culture and language developed within Ireland as the product of this interaction and mixing between various prehistoric populations.
The Milesian story could be a sort of ancestral memory of a connection between Bronze Age Iberia and Bronze Age Ireland (i.e., Beaker culture), or it could be something as simple as a way for the Irish nobility of the Middle Ages to legitimise their positions by tying themselves to mythological figures like Breogan and Bile, much as the Heraclids claimed to be the sons of Heracles or the Wuffings of East Anglia claimed to be the sons of both Woden (Odin) and Caser (Julius Caesar). We simply don't know.