With this observation in mind how far are we from defining $\forall \mathcal{C} \ \text{category}\ \pi_1(\mathcal{C})$? Let me be more clear.
Let be $n$ the following category $0 \rightarrow 1 \rightarrow 2 \rightarrow \cdots \rightarrow n-1$. We might define a path in $\mathcal{C}$ any functor $\alpha \colon n\ \to \mathcal{C}$ Given two paths $\alpha: n \to \mathcal{C}, \beta: m \to \mathcal{C}$ we might define the joint path $\alpha \star \gamma: n+m \to \mathcal{C}$ in the obvious way.
We cannot (i do believe) expect to have an inverse for a path. Now we have a notion of homotopy wich is given by natural transformation (maybe better natural equivalence?). And we can define.
$\mathcal{L}_c(\mathcal{C}):=\{\text{path who begins and end in }c \in Ob(\mathcal C) \}$
And then $\pi_1({\mathcal{C},c}):=\mathcal{L}_c$/homotopy equivalence.
I do believe this work have been done but I haven't found anything on catlab or wikipedia.
As I said in a comment I wouldn't call this construction the $\pi_1(\mathcal C,c)$.
First of all your construction distingushes between an $n$-tuple of composable arrows in $\mathcal C$ and their composite that in topological case is not considered: this problem arise because you consider $n$-tuples of composable arrows, i.e. a functor from $n$ in $\mathcal C$, quotiented up to natural equivalence, and we known that natural transformations are defined only between parallel functors, i.e. between functors with the same source and target.
No let's see more clearly what should be an equivalence class in the set you are considering. Given two paths $\alpha,\beta \colon n \to \mathcal C$ a natural equivalence $\gamma \colon \alpha \Rightarrow \beta$ between them should be given by a family of isomorphisms $\gamma_n \colon \alpha(n) \to \beta(n)$ such that obvious square commutes. That implies that for every $i=1,\dots,n-1$ the arrows $\alpha(i-1) \to \alpha(i)$ and $\beta(i-1) \to \beta(i)$ should be isomorphic in the arrow category. So an equivalence class in this case could be more likely be called a factorization class for a path, instead of a path, since such equivalence class retain up to isomorphism the factorization of an arrow.
It seems that what are you willing to do is to treat a category as a directed space and calculate the fundamental group of this space. I don't get the utility of this construction since the only way I see to continue on this road is a construction that would give you the group of automorphisms of an object quotiented up to isomorphism.
What can be a useful construction instead is to consider the nerve of the category $\mathcal C$ and the study its fundamental group (which is really similar to what is done for the study of the homology of groups).