Let's assume, for argument's sake, that the Galilean transformation holds rather than the Lorentz transformation.
Then your questions would become
Why is infinite speed arbitrarily the limit? Why is it that exactly
at infinite speed is where infinite energy is required to accelerate
any object with mass?
I suspect that you wouldn't, in fact, think of asking such questions since they almost answer themselves.
Moreover, infinite speed would be an invariant speed - a speed that is measured to be the same in all reference frames - since (loosely speaking) $\infty + v = \infty$.
If we ask the question "what if there is a finite invariant speed", the mathematical answer is the Lorentz transformation where $c$ is the finite invariant speed.
Indeed, if we let $c \rightarrow \infty$ in the Lorentz transformation, we recover the Galilean transformation.
From this perspective, the result that accelerating to the invariant speed requires infinite energy and is thus impossible, doesn't seem so odd.
To summarize, without postulating that the speed of light is invariant, one can derive the form of the Lorentz transformations from just the principle of relativity. In this form, there is an undetermined finite invariant speed.
That light propagates at the invariant speed is then simply an empirical fact rather than a postulate.