I can't resist posting my latest publication as an answer: Koenig, Külshammer, Ovsienko: Quasi-hereditary algebras, exact Borel subalgebras, A-infinity categories and boxes.
Let me sketch the result, which is exactly of the kind: You have a problem that is stated purely in down-to-earth language and the only known solution requires higher categorical language, in this case A-infinity categories and boxes.
The motivation comes from the classical example that in the representation theory of semisimple complex Lie algebras, one often studies induced representations. The universal highest weight modules, the Verma modules, for $U(\mathfrak{g})$ are induced from the simple modules for the subalgebra $U(\mathfrak{b})$, where $\mathfrak{b}$ is a Borel Lie subalgebra of $\mathfrak{g}$. The PBW theorem implies that $U(\mathfrak{g})$ is free over $U(\mathfrak{b})$, thus tensoring gives an exact functor $U(\mathfrak{b})-\operatorname{mod}\to U(\mathfrak{g})-\operatorname{mod}$ and one can study parts of the representation theory of $U(\mathfrak{g})$ by understanding the representation theory of $U(\mathfrak{b})$.
Now, the formulation of the problem is the analogue of this problem in the world of finite dimensional algebras. The analogue of $U(\mathfrak{g})$ is called a quasi-hereditary algebras. Roughly speaking, these are algebras having certain modules, called standard modules, which behave similarly to Verma modules for $U(\mathfrak{g})$. An "obvious" question is whether each quasi-hereditary algebra $A$ also possesses the analogue of a $U(\mathfrak{b})$, which is called an exact Borel subalgebra $B$. Here, the standard modules should be induced from the simple modules for $B$ and $A$ should be projective (slight generalisation of free) over $B$. This again implies that the induction functor, $B-\operatorname{mod}\to A-\operatorname{mod}$ is exact. Additionally, $B$ should satify an analogous property to solvability of $\mathfrak{b}$, called directedness. This formulation definitely no higher categorical methods and almost no category theory.
The answer to the question is no, if you want to stick to your algebra $A$. This is known since 1995. But if you allow to change from $A$ to an algebra which has the same representation theory, i.e. an algebra $R$ Morita equivalent to $A$, the answer is yes. So, also the answer to the problem requires no higher categorical methods.
In contrast, our proof does. We start by considering the Yoneda Ext-algebra of the standard modules, regarded as an $A_\infty$-category. We translate this to an algebra $B$ together with a $B$-coalgebra $W$ (this is what is sometimes called a box). Then the dual algebra of the coalgebra $W$ does the job. To prove that it does, in each translation step we also use different descriptions of the category of modules filtered by standard modules, i.e. twisted modules for $A_\infty$-categories, modules over a box.