Given a compact manifold $M$, I want to construct a finite collection of charts $\{\varphi_i\}$ such that $\varphi_i^{-1} \circ \varphi_j$ is bounded on where it is defined.
It seems like a simple minded problem, but the obstacle is that the domain of $\varphi_i^{-1} \circ \varphi_j$ is open, thus boundeness cannot be guaranteed naively. So my idea is to start with an infinite collection of charts then trying to shrink them so that all their intersection is precompact but you cannot really do that with an infinite collection.
Using your idea and assuming your definition of manifold is a Hausdorff, second-countable, paracompact topological space, then it has a metric that induces its topology. Using compactness, we can find a finite collection of charts $(U_i,\phi_i)^n_{i=1}$ for $M$. Each $p\in M $ is contained in a $V_{i,p}$ such that $\overline V_{i,p}\subseteq U_i.$ The collection $\left \{ V_{i,p} \right \}_{p\in M,\ 1\le i\le n}$ covers $M$ so there is a finite subcollection $\left \{ V'_{k} \right \}_{1\le k\le m}$ that also covers $M$. For each $1\le k\le m$, there is an integer $j_k$ such that $V'_k\subset \overline V'_k\subseteq U_{j_k.}$ This implies that $V_k'\cap V'_l\subseteq \overline V'_k\cap \overline V'_l\subseteq U_{j_k}\cap U_{j_l}$, from which it follows that the collection of maps $\left \{ \varphi_{j_k}^{-1} \circ \varphi_{j_l} \right \}$ is finite and bounded on $V_k'\cap V'_l.$