Is $C _v=Tleft(frac{partial S}{partial T}right)_{V}$ true irrespective of reversibility of the process?

The definition of $C_{V}$ is:

$$C_v=left(frac{partial U}{partial T}right)_vtag{1}$$

$C_{V}$ is independent on how the process actually goes. It's the heat capacity "if you had chosen a specific path where volume is constant". When we write $)v$, we are specifying a path, a path were the volume is constant. So there is no other path option. The actual path is irrelevant.

Furthermore, as $C_{V}$ is the ratio between two state functions ($U$ and $T$), it follows that it must also be a state function, path-independent.

We can then use relation you provided

$C_{V}=left(frac{partial Q}{partial T}right)_{V}=Tleft(frac{partial S}{partial T}right)_{V}$

which indeed is valid assuming reversibility. But since $C_{V}$ is path-independent, we can calculate using a hypothetical reversible path, which conforms to that equation. And the value will hold for any other irreversible situation. So the equation is correct, even in irreversible processes.

$C$ is path dependant. But $C_{V}$ is not.