The CMB originates from what's called the surface of last scattering. Prior to that time, density was such that photons couldn't travel very far without being absorbed. During that period, matter was in an ionized plasma state, consisting of free protons, neutrons, electrons, and some light ionized nuclei like deuterium, without any bound electrons. In this scenario, there were no objects bigger than a light nucleus. It would be impossible for stars or galaxies or even dust particles to form in that situation.
Reinterpretation would depend on what the evidence was. This isn't an area that we're particularly unsure of. There's a very comprehensive theory of the evolution of the early universe, supported by all the evidence we have for quantum theory. I recommend Steven Weinberg's book The First Three Minutes if you're interested.
When people still insist on believing the CMB proves it even when we now see fully formed galaxies at the edge of the observable universe, it's only because they're vested in that theory, and they know if it's wrong then they've been badly wrong for their entire lives. Only people with decades of vested belief in the Big Bang are hanging onto it for dear life, while open minded objective people correctly claim the evidence is against the Big Bang, and it was never anything but a wrong guess.