> So why would alien civilizations agree to such a thing instead of just adopting a planet and staying there?
I'm guessing you've not read them?
The gimmick of the series is that all sentient species were uplifted from non-sentient animals, and they were uplifted by species that were uplifted, which in turn were uplifted... and none of them knows by whom or by what. The "progenitors", the hypothesized first sentients that uplifted a presumed first generation, are lost to prehistory and nobody knows.
So there is a shared culture and a small degree of shared biology, because each generation of uplifters uses existing models of sentient species to model new sentient species. There are only so many ways to invent the wheel, although Brin was very imaginative indeed in this.
Humans were not uplifted; we're called "wolflings" and as a terrifying novel abomination would have been exterminated upon discovery, were it not for the novels' future setting having us already started work uplifting dolphins, chimpanzees, orang utans and a handful of other species. If humans were not a progenitor species the aliens would have extirpated us -- but we have client species of our own.
He does have answers to most of the objections one might come up with.
Brin's ideas were influenced by Larry Niven, and riffed upon by Terry Pratchett in his early SF works. Later Brin novels riff on ideas from his own earlier ones.
In their time, in the 1980s, I really enjoyed them. I own them all, several in hardback. I haven't re-read them in decades so I don't know how they hold up, but Brin is alive and still active.