Finding all solutions by Grover search(not superposition)
First note that the sum $O\left(\sqrt{\frac{N}{t}}+\sqrt{\frac{N-1}{t-1}}+\dots+\sqrt{\frac{N-t+1}{1}}\right) = O(\sqrt{Nt})$.

The quantum query complexity of this problem is indeed $\Theta(\sqrt{Nt})$.

The lower bound can be shown by reduction from the problem of deciding whether the input has $t$ marked elements or $t+1$ marked elements. This problem is very similar to $t$-threshold, and has a lower bound of $\Omega(\sqrt{Nt})$. This can be shown using the polynomial method or the adversary method.