Why $y[n] = x[-n]$ is not time-invariant?
A time-invariant system is one that, when you shift the input signal, the output is shifted by the same amount.

A system that reverses the signal cannot be time-invariant because when you shift the input, the output is shifted the other way. $k$ and $-k$ are not the same amount.

$$
y[n-k] = x[k-n] = x[-n \mathbf{+} k]
$$