Glagolitic very quickly got pushed out by what were essentially Greek letters. If you look at Bulgarian and Byzantine manuscripts from the time, they are almost impossible to tell apart, unless you know the languages.
The reason for that is pretty obvious if you look at the Glagolitic letters themselves: they are horrible UX. You need a lot more strokes than for something like Greek or Latin to record the same information. Because Glagolitic was contrived and not polished with use over the centuries, there was very little reason to use it over Greek.
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In reality, at the time, it was the Eastern Christian church that was more liberal, than Rome. Rome insisted every local church make services in Latin, and didn't translate it. The Eastern church instead, had the bible in Greek, but allowed to translate it in local languages. Initially, those translations were made with Greek letters, which weren't fully reflecting the phonology of Slavic and other languages, so they were extended, which produced Cyrillic.
As I understand, the same way Coptic script in Egypt, and Ge'ez in Ethiopia were made, thanks to Eastern Christian church allowing this.
Saint Cyril, in fact, invented the Glagolitic script, which was a development of cursive Greek. Cyrillic was named after him, and it was mostly Greek alphabet, plus some characters from Glagolitic, like Ⱎ, ⱍ and ⱑ.