In the top answer to this question (Is the Boltzmann constant really that important?) I read that the Boltzmann constant is just a dummy factor which converts energy to temperature.

But that allows me to put another numerical value in place of the Boltzmann constant but keep the dimension J/K.

I.e. what if, in $$S=c\ln W,$$ I put $c\approx 56 \, \mathrm J/\mathrm K$ in place of $c\approx 1.38\cdot 10^{-23} \, \mathrm J/\mathrm K$?

On this page (https://en.wikipedia.org/wiki/Thermodynamic_beta), the Boltzmann entropy using the Boltzmann constant implies thermodynamic beta, which implies (according to https://bouman.chem.georgetown.edu/S98/boltzmann/boltzmann.htm) the Boltzmann distribution.

So the Boltzmann distribution (from which one can derive the Maxwell–Boltzmann distribution and then the typical speeds of molecules) depends on the numerical value of Boltzmann constant. Then why is the Boltzmann constant just a dummy factor?

The typical speeds of molecules would be totally different if we used another numerical value (but the same dimemsion) of the above constant.