Property for time invariance of system in terms of its impulse response
Time-invariance means that if $y[n]$ is the response to $x[n]$, then $y[n-k]$ is the response to $x[n-k]$. So if $h_0[n]$ is the response to the impulse $\delta[n]$, then - if the system is time-invariant - its response to $\delta[n-k]$ must be $h_0[n-k]$.

If $h_k[n]=h_0[n]$ were true, then the response to any input $\delta[n-k]$ would be $h_0[n]$, regardless of $k$, which basically means that the system output would be independent of the input signal.