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The issue with informal principles like Mach’s is that they appeal to everyday language and as such are less precise than a mathematical theory; because, in Bohr’s words: “when it comes to fundamental physics, language can be used only as in poetry”.
Such (poetic) principles may help with motivating a mathematical theory but can become obsolete or even a conceptual burden once that mathematical theory is present. This is what happend to Mach’s principle and Newton’s bucket argument once GR was finally conceived.
Specifically on your argument: (1.) Dark matter is not needed to explain why stars are gravitationally bound in a galaxy, but is invoked to explain their peculiar observed velocity distribution as a function of their distance from the center. (2.) If it were true that in a reference frame there is “no centrifugal force” then galaxies would collapse under the remaining force of gravitational attraction.
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