Goods could still be sent on foreign ships from US ports to Europe. There was no requirement that the goods be sent to Britain or Europe on US or Northern ships, and no requirement that goods sail out of New York or Boston, rather than Charleston or New Orleans.
It is still the case (per the Merchant Marine Act of 1920) that cargo shipping between US ports sail on US ships. That hasn't prevented Louisiana and Texas from having the busiest US ports in terms of tonnage, but Los Angeles - Long Beach and New York - New Jersey are still the biggest ports in dollar terms. Today or in the 1850s having a large population means having a major port, and the ports where imports come in isn't automatically the same place where exports go out.
The Warehousing Act of 1846 also gets blamed for New York's predominance. That law said that importers didn't have to pay tariffs immediately on imports. The goods could be stored in warehouses until they had the money available. Nothing required that goods go through New York. It was just that New York built many warehouses.
New Orleans did too, but it didn't have the resources on hand, and also had to worry about flooding, drainage, epidemics and other problems. Those problems would have remained even without the Warehousing Act.
One difficulty with all of the theories is that it's not always easy to tell a Northern from a Southern from a British firm. Early coastal shipping in Texas was dominated by a Northerner, but whether his firm represented New York City rather than the ports it served in the South is something that ought to at least be discussed. Agents in New Orleans made most of the everyday decisions. Charleston had direct cargo shipping to Britain, but whether the firm counted as Northern or Southern or British could only be sorted out when war forced the owners to choose a side.
Everything you wrote made sense. I don’t get this idea that the Navigation Act of 1817 was part of some Yankee conspiracy to enslave the South. I’m doing a lot of research and it reflects part of what you alluded to. Companies weren’t all “Southern” or a “Northern” or even “British.” There was a whole lot of synergy going on, just like today.