Why do you even need a Navy if you're not coastal or vulnerable? Only reason I can think of is to build carriers for nukes. True, having a Navy makes it a lot easier to conquer civs on other continents, but it also makes YOU easier to conquer by civs on other continents.
Even on pangaea a navy is usually useful. Just being able to put out a caravel is worth it! But you are questioning utility for continents? Madness! Your coastal spots should not have more than one water hex, and the AI is terrible at invasions. If anything, your coastal cities are more secure than the inland ones!
There can be a barb camp ten tiles from your borders that's on the coast, and it will barb galleys into your territory. Your trade route doesn't have to be in the fog to get pillaged by barbs.
Yes, it happens. But asserting you need 2-3 triremes to counter is silly when the city+archer deals with any barb problems for the vast majority of games. The main reason I don't bother building triremes is because I have a hard time leveling them up against barbs -- there are not enough of them for my tastes!
Okay, so why did you raise it as a problem?
I'm saying what's the point of settling coastal if you're not going to build cargo ships or take advantage of sea resources?
Obviously you will want to be building cargo ships, but they don't have to be particularly early. You want to attract trade routes too, open trade with multiple AIs, and have more options for CS trade route quests. Sea resources are a nice bonus, but not required to justify settling the coast.
I had only 2 Iron on my ENTIRE continent. A guy frig rushed me and I could only have 2 frigates at a time because of this.
You mentioned Aztec player moving to the coast. So is the context of your question more about MP than SP? I would see that as a very different set of circumstances! Two frigates defending a city is plenty though. One in the bay, one in the city.
Luckily my cap was in an extremely defensive spot, tucked away in a coastal pocket surrounded by land so he couldn't get to me with more than 2 frigs or 1 frig/1 melee at a time. But if I wasn't so lucky I would have been screwed.
One water hex is usually all you have to worry about, so you just needed average luck. One melee ship per turn is all you have to face. The AI is hardly ever a problem, even with how they spam units. With a land attack, the AI will be throwing units in the water, so your coastal city is even better defended.
When you settle coastal, are you not taking a bet that you will have access to Iron later on?
It is not like iron is rare, but no. Building coastal cities does not require building a frigate-oriented navy (or even much of a navy at all). In your example above, a privateer in your bay might have been enough to let you laugh at the attack rather than sweating bullets. Certainly an Ironclad would have done it!