What made Roman script suited for adoption?
The fact it was adopted a lot. Latin on its own is not particularly suited for a lot of phonemes, but it was the only game in town in Western and Central Europe, and that meant there was a long, long tradition of workarounds—both digraphs and diacritics. So another digraph or diacritic for another language was not a big deal. Only Vietnamese really stretched that.
Ultimately, hegemony came first: it was the script of the Catholic Church and the mediaeval intelligentsia—if you want to write down your barbaric dialect, deal with it.
What made Greek script suited?
Hegemony of Greek culture and the Orthodox Church, meaning it was a target for Bactrian and Cyrillic, and Balkan languages and Turkish. On Cyrillic see below.
But it wasn’t repurposed that often, the languages it was repurposed for were usually not mainstream languages (or it was not the mainstream spelling of a mainstream language). And while there are traditions of both digraphs and diacritics in Greek, they have never become mainstream themselves: a little digraph work in Greek dialect (Cypriot, Tsakonian), diacritics limited to Greek dialectology. That means that it was not a very good fit for other languages most of the time.
What made Cyrillic script suited?
Hegemony of the Russian Empire and the Soviet Communist Church. There was a merry-go-round of scripts from Arabic to Latin to Cyrillic to Latin.
A broader phonetic repertoire than Greek or Latin, which helped.
The broader phonetic repertoire was because Cyril was not afraid to add new letters. Greek did it only once or twice. Outside of the IPA, and the IPA-inspired African alphabets, Latin only did it once or twice. Through the precedence of Glagolitic, Cyril added a dozen.
And Cyrillic kept adding new letters (typically as variants of old letters) whenever a new language’s inventory showed up; there’s more new letters than digraphs and diacritics (though you get those too). Just as well they did: they have the languages of the Caucasus to deal with.
Why did they keep adding new letters? As with Roman and diacritics: precedent. That’s how Cyrillic, as distinct from Roman, started adapting to new languages.
What made Arabic script suited?
An OK consonant repertoire, and diacritics meaning that you can add more consonants as variants (three dots for Farsi, four dots for Shina language, as well as digraphs for Urdu). OTOH, avoiding vowels outside of matres lectionis. Doesn’t work well outside Semitic.
And Hegemony of Islam.
What made Hebrew script suited?
I don’t know if Hegemony and Judaism go together, but that. And as for Arabic: some diacritics and digraphs, though not common enough outside of Yiddish to be a fine-tuned instrument; and only matres lectionis for vowels.
Don’t know enough about other scripts.