$\DeclareMathOperator\colim{colim}\DeclareMathOperator\CS{CS}\DeclareMathOperator\CSN{CSN}$The examples considered in algebraic topology are usually like $\mathbb{R}^\infty=\colim_n\mathbb{R}^n$ or $O(\infty)=\colim_n O(n)$ (the infinite orthogonal group) or infinite-dimensional CW complexes.  These can usually be written as the colimit of locally compact Hausdorff spaces $X_n$ and
closed inclusions $X_n\to X_{n+1}$, where $X_n$ typically has
empty interior in $X_{n+1}$.  In this context, a standard lemma
says that every compact subset $K\subseteq X$ is contained in
some $X_n$ and so has empty interior in $X_{n+1}$ (and therefore
also has empty interior in $X$).  Thus, examples of this type
will never be locally compact.

One could consider the class $\mathcal{C}$ of $k$-spaces in which every compact set has empty interior, which is a kind of opposite to local compactness.   I think that this class of spaces contains most of the popular examples and has good closure properties.

On the other hand, very many spaces can be written as a quotient of an LCS space.  Indeed, let $\CS(X)$ be the set of continuous maps $\mathbb{N}\cup\{\infty\}\to X$ (i.e. pairs consisting of a convergent sequence and its limit).  We can give $\CS(X)$ the discrete topology and the space $\CSN(X)=\CS(X)\times(\mathbb{N}\cup\{\infty\})$ the product topology.  This makes $\CSN(X)$ into an LCS space, with an evident surjective evaluation map $\epsilon\colon \CSN(X)\to X$.  The space $X$ is said to be *sequential* if $\epsilon$ is a quotient map.  (The definition is usually phrased differently, but easily seen to be equivalent.)  One can check that if $X$ is a $k$-space and every compact subspace is metrisable then $X$ is sequential.  This covers most examples typically considered in algebraic topology.

My guess is that the space $\beta(\mathbb{N})$ (the Stone–Čech compactification of the naturals) is not a quotient of an LCS space, but I have not checked that.