I would like to understand if there is deeper reason/motivation behind augmentations in homological algebra. Recall classically in homology if there is a complex of free $R$-modules ($R commutative unitary ring)

$$ ... \rightarrow{\partial_{n+1}} C_n \rightarrow{\partial_{n}} C_{n-1} \rightarrow{\partial_{n-1}} ... \rightarrow{\partial_{2}} C_1 \rightarrow{\partial_{1}} C_0 \to 0 $$

then it can be turned to augmented chain complex by extending the right part to

$$ ... \rightarrow{\partial_{2}} C_1 \rightarrow{\partial_{1}} C_0 \rightarrow{\epsilon} R \to 0 $$

where $\epsilon(\sum_i a_i \sigma_i)):= \sum_i a_i \in R$

such that the $0$-th homology $H_0$ of the original complex and of the augmented complex $H^0_a$ are related by $H^0= H^0_a \oplus R$.

Fine, but the question is if there is any deeper philosophical reason behind using augmentations or is it just a historical leftover? Moreover was it to use this concept just a matter of "taste" or was there mathematical reason?

(Maybe it's appeared in certain sence more flexible for certain generalization directions. For example to work in some situations more "uniformly", I don't know, that's just a guess) Obviously there is no information gain/loss when passing to augmented objects. Are there nevertheless any advantages to pass to them? For example does this concept maybe provide better structural /functorial compatibilities or say "more natural framework" when dealing in more general setting (eg simplicial objects in higher categoy theory)?