Classes of languages with objective-independent memory

We start from the following fact: Given a two-player game arena $G$ labelled with an alphabet $\Sigma$, and an objective $W \subseteq \Sigma^\omega$ for Player 1, if $W$ is closed by subwords then if Player 1 has a winning strategy, she has one using memory $\leq |G|!$. The remarkable thing is that this memory does not depend on $W$, just on $G$. We say that a class of languages $\mathcal{C}$ has the $\textit{objective-independent memory property}$ if there is a function $f$ such that for all arena $G$ and $W \in C$, if Player 1 has a winning strategy for $W$ in $G$ then she has one using memory $\leq f(|G|)$. This property is useful in distributed synthesis. As we saw above, the class of subword-closed languages has this property, with $f$ the factorial function. The problem is to characterise classes of languages with this property, or at least identify some non-trivial examples. A first step could be to show that for all $k$, the class $C_k$ of languages recognised by automata with SCCs of size at most $k$ has this property, and identify some classes that have that property but are not subsumed by some $C_k$.