Bronze continued to be the prestige metal. The advantage of iron was that it was plentiful and cheap. I’m no expert, but I gather that bronze was actually superior for weapons and armor, but was too costly for mass use. Eventually, of course, steel trumped both bronze and iron.
Attempt at humor. Iron won out over bronze for its ability to hold an edge, especially after they learned to forge with some carbon to produce steel. For other applications, bronze had the advantages that you mention.
The pinnacle of iron was the US Civil War, although I believe the Eiffel Tower was constructed with very low carbon iron.
You are correct, iron was plentiful, cheap and could be worked by simpler methods.