Well, let's look at the word "know" in that explanation.
First, it is assumed that the cosmic microwave background radiation is black body radiation that has been stretched by cosmic expansion.
It is assumed that the "black body' that emitted that radiation is only hot hydrogen gas/plasma condensing to monatomic hydrogen and hydrogen-hydrogen molecules.
And, it is assumed that the rate of cosmic inflation is the Hubble Constant, which now doesn't seem to be a constant.
So, if we can't really calculate how pure hydrogen gas would condense and collapse to make stars in the "early universe", how do we really know what the CMBR would look like from first principles? Isn't this potentially a lot of confirmation bias by folks with at theory who want to support that theory with the discovery of the CMBR?
The problem is that the farther back we can look, the more the theoretical time line does not seem to be matched by the new observations. So, it seems rational to question the time line in the theory. But, the tendency of the theorists is to stick with the time line and propose "new physics", instead. Which raises red flags for the theory in the minds of those of us whose minds are not already committed to the BBT as the only possible "truth".
The advancement of science requires people who can "think outside the box", even in cases where the correct solution is eventually found to lie inside the box. To me, the BBT is being given too much respect as the box containing the truth, somehow.