Put another way, it actually is important here that the set of colors we are talking about is finite. It's NOT the case that just having every finite subgraph b

Put another way, it actually is important here that the set of colors we are talking about is finite. It's NOT the case that just having every finite subgraph be C-colorable entails that the entire graph is C-colorable, when C is infinite. For example, if G is the complete graph on an uncountable set of nodes and C is a countably infinite set, then every finite subgraph of G is C-colorable, but G itself is not. So somewhere in the argument, the finiteness of C must play a role. This is in ensuring compactness.