Trade is a huge income for me in the game. However, I heard that province that benefit from trade has to be connected to your capital by direct land contact. Tosa's capital is on a island and i noticed that only the four provinces (with my capital) have trade routes. So does that mean upgrading port to trade ports and spending on trade boasting buildings are useless in provinces that are outside of that island?

I have spent a lot of time and money on upgrading the ports and infrastructure of other provinces that are outside of the island - like trading ports and finance buildings criminal den etc.

list of my situation:
1.only the foreign powers and three partners to trade with
2.provinces conquered that is away from my tosa island
3.only physical connection between tosa island and other provinces possible are the straits between tosa island, Awaji and Harima (Harima and Awaji are control by me)
4.while a handful of others are not connected at all - Tsushima, Goto etc.

I checked those provinces, the commerce income is larger in proportion to farming, so does it mean the trade buildings and ports help? or was it the industrial buildings that works?