Following @JamieGabe's [hint](https://mathoverflow.net/questions/407455/c-sum-i-a-ii-ge-a-ij-for-matrices-with-entries-in-a-c-algeb#comment1044862_407455), we claim that 

$$\lVert(a_{i,j})\Vert \le n \Bigl\lVert\sum_i a_{i,i}\Bigr\rVert.$$

Indeed, the map

$$\operatorname{Tr}: M_n(A) \to A: (a_{i,j})\mapsto \sum_i a_{i,i}$$
is easily checked to be completely positive. 

Let $\{u_i\}$ be an approximate unit for $A$. Then the matrices
$$U_i:= \begin{pmatrix}u_i & 0 & \cdots & 0\\ 0 & u_i & \cdots & 0\\
\vdots & \vdots & \ddots & \vdots\\ 0 & 0 &\cdots & u_i\end{pmatrix} \in M_n(A)$$
determine an approximate unit for $M_n(A)$. By general properties of completely positive maps,
$$\lVert\operatorname{Tr}\rVert = \lim_i \lVert\operatorname{Tr}(U_i)\rVert = \lim_i  \lVert\nu_i\rVert = n$$
and the desired inequality follows.