Why do researchers only count the number of multiplications when analyse the time complexity of Matrix Multiplication?
It looks like the article was written by someone who does not understand matrix multiplication.

the number of additions is equal to the number of entries in the matrix, so four for the two-by-two matrices and 16 for the four-by-four matrices.

With the classic matrix multiplication algorithm (which is the one explained in the example) between two $4\times 4$ matrices, each coefficient of the product requires $3$ additions, so a total of $3\times 16 = 48$ additions, not $16$.
Usually, when it is said that only multiplications matter and not additions, it means of matrices, not coefficients.
For example, a &quot;naive&quot; divide-and-conquer strategy compute $A\times B$ where $A,B$ are matrices of size $n\times n$ by doing $8$ products of matrices of size $\frac{n}{2}\times\frac{n}{2}$ (and some additions of matrices, but those are done in complexity $O(n^2)$, which is negligible in front of the complexity of matrix multiplication). That way, the complexity verifies $C(n) = 8C\left(\frac{n}{2}\right) + O(n^2)$ and it is easily proven that $C(n) = O(n^3)$, so this strategy is not an improvement.
The Strassen algorithm use a divide-and-conquer strategy to improve complexity: it does $7$ products of matrices of size $\frac{n}{2}\times\frac{n}{2}$ and some additions, so the complexity verifies $C(n) = 7C\left(\frac{n}{2}\right) + O(n^2)$ and we get $C(n)\simeq O(n^{2.8})$.
In the two examples above, the number of matrices multiplications matter much more than the number of matrices additions. But in both, the number of multiplications/additions on coefficients is not compared.
It is confirmed in the article:

Volker Strassen reportedly set out to prove that there was no way to multiply two-by-two matrices using fewer than eight multiplications. Apparently he couldn’t find the proof, and after a while he realized why: There’s actually a way to do it with seven!