In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 281 line 17-line 18:
we have a fibre bundle $M_\infty\to (M_\infty)_M\to BM$ with $(M_\infty)_M$ constractible. In order to prove Proposition~1 (Group-Completion Theorem), is it necessary to apply Proposition~2 of the paper Homology fibrations and group completion theorem, McDuff-Segal? I noted that by applying Proposition 4.66, Algebraic topology, A. Hatcher, I obtain that $\Omega BM$ is weak homotopy equivalent to $M_\infty$. Hence we do not need Proposition~2 of the paper Homology fibrations and group completion theorem, McDuff-Segal anymore. Is it true?
$M_\infty\to (M_\infty)_M\to BM$ is not always a fiber bundle, not even a (Serre-)fibration. It is a homology fibration, which is a weaker notion.There will be no long exact sequence of homotopy groups, so you can't argue as in Hatcher's book.
Let me give you a general advise based on the type of questions you ask these days. I strongly recommend you to read Allen Hatcher's account on the group completion theorem in Appendix D of his exposition of the Madsen-Weiss theorem, before trying to fully understand the fantastic six pages of McDuff and Segal. Please notice that there are known some small mistakes in the paper, some of them mentioned here. Compare Martin Palmer and Jeremy Miller's recent account of the group completion theorem, which does not have these issues.