Is it bad to own most of the provinces with ports

I am right in thinking this?

Isn't it true that you only trade with ports owned by other provinces?

So if you owned most of them you would get very little trade income.

Is that right?

Right - you only trade with ports that you don't own AND ARE NOT AT WAR WITH THE OWNER.
Since trade income is required to get rich quick, this limit drives most of my grand strategy - what direction to attack in and who to pick wars with. For example, when playing as the Turks, I first capture the rich trading ports in the Eastern Med (Egypt, Antioch, Tripoli, Constantinople, Khazar). Then I build out my trade fleet over the rest of the Med and beyond, avoiding wars with the nations bordering the med - Spanish; Alms; Italians; Sicilians. Then I conquer the map from the inside out - Taking out Golden Horde, Novrograd, Poles, Hungarians, HRE - with minimal disruption to trade income.
Finally for the coup de grasse, amass a force capable of capturing the rest of the map as rapidly as possible.
There was a great post about how this trade dynamic creates an endgame problem - when you have most of the map, you are haemorraging cash, so you need to have a huge cash stockpile to ensure you can still build replacements during the final push.

Of course, you can do what I do now and play without trade, since this makes the strategic game too easy. That way you face the old STW trade-offs with not enough cash to go round