Sorry all, but the impedance of free space is set by the choice of units. If you use natural or Planck units where $c$ and $\hbar$ and other constants are 1, the Planck impedance comes out to be about 30. More to the point, Planck units sets Coulomb's constant $\frac{1}{4 \pi \epsilon_0} = 1$. This represents a choice of units.
To the point. If instead you set the permitivity $\epsilon_0 = 1$, along with $c=1$, then the permeability $\mu_0$ is also 1, and the impedance of free space is also 1.
It is simply units, there is no physics behind the 377 ohms.
See Wikipedia at https://en.m.wikipedia.org/wiki/Planck_units
