I guess the answer to my question is as follows.
If $A$ is a finite dimensional Banach algebra with a trivial Jacobson radical, then it is amenable. The answer is easily based on this fact that this class of finite dimensional algebras are isomorphic to a product of finitely many $n_i\times n_i$ matrix rings over division rings $D_i$, by Artin–Wedderburn theorem.
Recall that each of this matrix algebras is amenable and a simple argument regarding adjoining the (virtual) diagonals together implies that the product is amenable.
The converse is also true. If $A$ is a finite dimensional Banach algebra with a non-trivial radical, it cannot admit any (virtual) diagonal; hence, it cannot be amenable.