But we play this "we have to call them what they call themselves in 'their' language' game selectively.
If you are or claim to be oppressed, then you dictate what you're called. Muslim vs Moslem, for example. Roma. Etc. We are even to switch phonologies for Latino names--lose the strong stress, modify vowels, pronounce the /r/ correctly, etc. Otherwise we can be condemned. You have to be among those of the right SES or politics for them to suddenly care about imposing non-English phonology or even morphology on monolingual English speakers.
If you're not oppressed or simply don't care to wield that cudgel, or nobody cares about your claim because empathy stretches only so far, there's no pressure to do this. Nobody calls Russia Rossiya, lots of people who are otherwise left of center will still say "Czechoslovakian" and when pronouncing the PRC's leader's name nobody cares to even try to get the phonology right. Navratilova and other Czechs (or Slovaks) have their names mangled routinely, no concern about palatal stops or vowel length. Never heard a Russian insist on palatalizing consonants in their names or even worrying about the expression of gender. Meh.
But we get bent out of shape over "Pakistan" with a fronted /a/, ignoring the phonemic length that English speakers are fully capable of.
As for "Muslim," I've known more than one who got bent out of shape because that /s/ after a stressed syllable is sometimes pronounced voiced, regardless of orthography. As though we're all taught that one specific Arabic lexeme. (Note that in colloquial varieties of Arabic, short u and short i are often lowered to o and e.) And in Texas and other places in the US where the i ~ e distinction before nasals is neutralized it's an especially humorous thing to insist on.
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