You could define the Big Bang to be at any value of t, i.e., pick t0 > 0. Such a shift changes nothing and won’t fix any singularities in the dynamics.
The problem is the absurdity of the idea that `t` could ever cease to be a linear value. Causality is a fundamental truth. To lose the idea of "before" and "after" is to lose causality.
If your fundamental theorem of the universe requires you to forego causality, you should consider looking elsewhere.