I would just like a clarification related to closed subschemes.

If $(X,{\cal O}_X)$ is a locally ringed space and $A\subset X$ is any subset with the subspace topology then $i^{-1}{\cal O}_X$ will be a sheaf of rings on $A$ where $i:A\rightarrow X$ is the inclusion map. (Recall that the inverse image $i^{-1}{\cal O}_X$ is the sheafification of the presheaf $U \mapsto \lim_{V\supset U} {\cal O}_X(V)$ on $A$, where the projective limit is over all open subsets $V$ of $X$ containing $U$.)

Is the reason why we don't do this (and instead start talking about closed subschemes, etc. etc.) just that $(A,i^{-1}{\cal O}_X)$ need not be a scheme even when $X$ is?

Put differently: given any closed subset of a scheme there will be many ways to make it a closed subscheme. What is the relation between the locally ringed spaces on a closed subset making it a closed subscheme and the locally ringed space I have described above which we obtain by pulling back the structure sheaf via the inclusion map.

I would just like a clarification related to closed subschemes.

If $(X,{\cal O}_X)$ is a locally ringed space and $A\subset X$ is any subset with the subspace topology then $i^{-1}{\cal O}_X$ will be a sheaf of rings on $A$ where $i:A\rightarrow X$ is the inclusion map. (Recall that the inverse image $i^{-1}{\cal O}_X$ is the sheafification of the presheaf $U \mapsto \lim_{V\supset i(U)} {\cal O}_X(V)$ for $U\subseteq A$ open, where the inductive limit is over all open subsets $V$ of $X$ containing $U$.)

Is the reason why we don't do this (and instead start talking about closed subschemes, etc. etc.) just that $(A,i^{-1}{\cal O}_X)$ need not be a scheme even when $X$ is?

Put differently: given any closed subset of a scheme there will be many ways to make it a closed subscheme. What is the relation between the locally ringed spaces on a closed subset making it a closed subscheme and the locally ringed space I have described above which we obtain by pulling back the structure sheaf via the inclusion map.