With any text in a Latin alphabet, in any language, show me the phonology of the language and I can read it aloud at normal speed with decent pronunciation within a few hours. If it’s in a different writing system, it’s going to take me days or longer to get up to even a moderate reading speed—and a lot of that would be cheating by just memorising the text and using the writing as a cue.
I remember an Irish guy who was speaking pretty fluent Italian mispronouncing the word "firma" making it sound like English "firm" instead of "feerma", and I'm pretty sure that wouldn't happen if he had to learn a new alphabet. As part of learning the new alphabet he would match that gliphy with the "ee" sound with less chances of associating it with the English sounds just because of similar spelling.
Fortunately that was a minority—maybe 5–10% of students had that difficulty and couldn’t grow out of it. Part of the issue is also that people seem to hold back and don’t try to really imitate the accent of a language, which they may not get is really an essential part of the pronunciation. You’ve got to put on your best “French accent” if you want to properly speak French!
Speaking of “closest sound in your native language”, my Polish coworkers and I a couple of jobs ago had a lot of fun one day when I transcribed their English speech using Polish spelling—lo and behold, it gave quite an accurate representation of their accents. (E.g., “It would be funny” = “it łód bi fany”.)
Latin alphabet might give you the illusion that you can read a text aloud but it's really just that, an illusion. Ask a French speaker with no knowledge of English to read this comment aloud "phonetically" and see if you can even recognize the words.
It's even conceivable that using a different alphabet actually helps you getting the pronunciation right since you might be less tempted to read it in English.
As a practical joke, I once asked a French speaker (who spoke English well) to read this aloud:
Un petit d'un petit
S'étonne aux Halles
Un petit d'un petit
Ah! degrés te fallent
Indolent qui ne sort cesse
Indolent qui ne se mène
Qu'importe un petit
Tout gai de Reguennes.
He was the only person in the room who didn't understand what he was saying. Indolent qui ne sort cesse
indolent = and all 'equi ne s- = king's
-ort cesse = 'orses
In comparison I was able to learn Korean alphabet quite easily.