It depends on your reference point…

What exactly does /məsjø/ mean to you? If you mean that you were trying to use a sound like the one in the last syllable of the English word “comma”, or in the first syllable of the English word “manipulate”, then don’t do that. The English sounds transcribed as /ə/ are not very close to the French sounds transcribed as /ə/. Neither is necessary exactly the same as the IPA reference point for the symbol [ə].

When enclosed by slashes, /ə/ is a “phonemic” symbol. Different languages have different pronunciations associated with their phonemes, even if the phonemes happen to be transcribed the same way. (Another example of this: French /t/ is not the same as English /t/.)

Based on my understanding, the pronunciation of monsieur in a Parisian accent is not significantly different from the pronunciation of /møsjø/. There is a definitely an association between the sound of schwa and the sound /ø/ in accents of that type: for example, the schwa in le is replaced by /ø/ when it is suffixed to a verb in the positive imperative form, as in fais-le. You can see more details in Maroon’s answer to Pronunciation of E in je, le, ce, ne, que. The sound /œ/ (which only contrasts with /ø/ in limited contexts) is supposed to be another possible realization of schwa.

But in other contexts, schwa may be subject to elision. In particular, this behavior has an important effect on the pronunciation of many monosyllabic function words that end in /ə/. Because French “schwa” shows this special behavior, it is represented by a special symbol /ə/ in IPA, even though the sound used to realize this phoneme can overlap with the sound used to realize the phoneme /ø/.

Another reason for the use of /ə/ is just conservatism. Because of the history of where the IPA was developed, French and English have fairly long histories of being transcribed into the IPA. Although it might seem counter to the original goal of transcription, IPA transcriptions can become conventionalized, and when pronunciation changes but a transcription does not, you get mismatches between the symbol used for a phoneme and the phoneme’s actual phonetic realization. A notorious case in French is the representation of nasal vowels, especially /ɛ̃/, which often has a more open pronunciation than that transcription implies.

Depending on the dialect of the speaker, schwa might be realised as [əʷ], [œ] or [ø].

Whatever this realisation might be, it’s still its own phoneme, since it has a very different behaviour from /ø/:

• /ø/ can appear in both open and closed syllables for most speakers, while speakers who realise /ə/ as [ø] in open syllables (as your speaker did for monsieur) normally pronounce it [œ] in closed syllables as in “je le dit” pronounced [ʒœl.d̪i]

• /ə/ is also unstable in ways /ø/ isn’t, disappearing in certain contexts.

• /ə/ can appear at the end of a word without attracting stress (at least in certain dialects) while /ø/ can’t.

So there are good reasons to transcribe /ə/ and /ø/ (and /œ/) differently. /ə/ has the advantage of conservatism being the historical transcription. It also provides a very distinct symbol for the dialects that clearly distinguish schwa from /œ/ and /ø/ and is a symbol used for other weak or unstable vowels in neighbouring languages.

And there is of course no rule that a phonemic symbol has to correspond to its phonetic realisation, although it helps. Like orthography, IPA transcriptions tend to drift with time.

I’ll add that despite what I just wrote, the similarity in realisation between /ə/, /ø/ and /œ/ leads to some interesting questions about the transcription of words like pâquerette, céleri (in Belgian French), the object pronoun -le, or “ce” in the expressions “et ce” and “sur ce” that have stable schwas that are always going to be pronounced and attract stress like any other vowel would. In the dialects that pronounce those schwas [ø] or [œ], shouldn’t we transcribe them /ø/ or /œ/?

Likewise with a historical /ø/ that has become unstable like that of “peut-être”, maybe a transcription as /ə/ would be more suitable. (This is especially true in Belgian and Canadian French, which lower the ⟨eu⟩ of peut-être to [œ], which is the normal realisation of schwa in those dialects)