Why can't the constancy of the speed of light be deduced from classical physics?

The kinetic energy growth is asymptotic, meaning it approaches infinity as the velocity approaches some value.

Unfortunately, this result already assumes that you know that there is an invariant speed. Without the invariant speed the formula for KE is $KE=\frac{1}{2}mv^2$ which has no limiting speed. It is only after you already know about the invariant speed that you get the expression $KE=((1-v^2/c^2)^{-1/2}-1)mc^2$ which goes to infinity as $v$ approaches $c$.

photons don't have mass so they can

This also requires already knowing about the invariant speed. Without the invariant speed there is no known relationship between all three of mass, energy, and momentum. With the invariant speed we learn $m^2 c^2=E^2/c^2-p^2$ which given the energy and momentum of light implies that light is massless.

So yes, those things, if known independently somehow, would have led to the conclusion of an invariant speed. But how could they have been known? Perhaps they could have simply been measured and known experimentally first, but historically it didn’t happen that way. Historically the invariance of c was postulated prior to measurements experimentally showing those points. Furthermore, such measurements would have been considered violations of classical physics.

In classical physics the kinetic energy of an object is $$E_{\text{kin}}=\frac{1}{2}mv^2,$$ which clearly has no asymptotes. That is, the second step in you argument is wrong.

In special relativity the kinetic energy is $$E_{\text{kin}}= \left(\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^2}}-1\right)mc^2,$$
which does have an asymptote. However, this formula is derived with the constancy of $c$ as a starting assumption.

In classical mechanics, particles can have arbitrarily high speeds and still have finite energies, so your statement 2 fails. Given any finite speed $v$, the energy is $1/2 mv^2$. We can keep on increasing $v$ and the energy will increase and still remain finite.

Suppose you think the maximum speed is $C$. Then a particle with such a speed has energy $1/2mC^2$ (which is finite). Why can't a particle have speed $C + \delta C$, which has energy $1/2 m(C+ \delta C)^2$, which is still finite?