In comments, John Lawler answered:
No, some schwas are phonemic in English. The stressed [ʌ] in butt and stump is a phonemic schwa. There is no contrast among central vowels, so all of [ʌ], [ə], [ɜ], and [ɨ], among many others, occur as allophones of the phoneme /ə/. Those schwas may be spelled with any vowel, just like reduced schwas; however, they're much less frequent, since any unstressed syllable is reduced, but phonemic schwas don't occur more frequently in lexical items than any other phonemic vowel. That means that most schwas in polysyllabic words are reduced, but this isn't true in short words.
[Not phonetically, but phonemically.] Stressed /ə/ has an [ʌ] allophone; they're never in contrast and they're phonetically similar; that's the definition of two allophones of the same phoneme. As I said, there's no contrast with English central vowels, so /ə/ assimilates to any environment, just like /h/ does.