Yes, but invoking Zorn's lemma on that poset does nothing useful. This just proves the trivial fact that the big graph exists. How do you prove that a coloring of the big graph exists?

And if we take the poset to be partial colorings ordered by inclusion, it's not the case that a maximal partial coloring must color the entire graph. Some partial colorings have made choices which prevent them from being extended any further.

[To be clear, whenever I say "coloring", I mean from a fixed set of colors. E.g., a 4-coloring.]

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.