The characters' stat blocks may not increase but the gold does increase the characters' effectiveness through more control of the narrative. To use secondhander's and Lancelot's players as examples the ships and pegasi they now own give them more freedom and options for traveling. The ship captain and owner of the shipbuilding business will have more influence on the community and have hirelings at their disposal. The monk will wield considerable influence as the Grand Master for an entire order of monks. The money certainly increases the effectiveness of these characters, even if their stats remain the same.
And yet we all know that in the typical campaign the characters will be able to go to where they are needed without the ships or pegasi, too or that all adventures they go on can be solved without additional influence and hirelings.