Docks are in the City Center, so the Cothon could essentially just replace the City Center. It would essentially be a City Center with additional Harbor traits which allows you to construct both City Center and Harbor buildings in it. Phoenicians would be barred from constructing Harbor districts, and they must settle on Coast/Lake to gain the Cothon (or possibly be completely limited to Coast/Lake).
This would solve the inland Harbor redundancy issue, the inland Harbor issue, the conversion issue, the canal issue (Cothons weren't canals but city centers can act as such so it would be a 2 in 1), while also providing an ability reminiscent of that of Civilization 5 (Free Harbor in the city) and uniquely make them very dependent on coast.
It would ALSO directly allow sea trade routes without having to research Shipbuilding.
Harbor adjacencies would then transfer to the City Center (so no clashing here) while possibly adding more adjacencies to Industrial, Campus, and Holy Site districts.
It makes so much sense!
Another hint (although this is VERY much a reach) is their discussion of Valetta in particular in the Ottoman stream which they frankly randomly brought up. Valetta allows you to purchase city center buildings with Faith. Valetta is in Malta, an island colonized by Phoenicians. See where I'm going with this? The synergy would be real.
How about this:
All Phoenician Cities have to be founded on the coast or no more than 1 tile away from the coast.
As soon as the city is founded, it gets a Free District: The Cotton, placed on the nearest coastal waters tile and, if necessary, providing a 'canal' -like connection to the City Center one tile inland.
This solves the problem of trying to put a City Center on water, which would be required if the Cothon = City Center.
It also gives a little bit of flexibility to the City Center placement while still making Sea Trade/Coastal positioning one of the Civ Attributes.
It makes Sea Trade Routes possible to all Phoenician Cities just as soon as they want to allow it.
It actually increases the adjacency bonuses from the Cothon, since there is now a possibility of getting adjacencies from 2 different tiles - one on land, one next to the land.
This also, coincidentally, gives the Phoenicians an advantage in Canaling, since they potentially could already have a 2 - tile (Cothon + City Center) 'canal'. That would allow them to take advantage of two-tile isthmuses as city sites with extra sea trade possibilities and more efficient naval movement.
Throw in a Unique Phoenician Sea Trader with longer range, more Gold Bonus (and maybe a Science Bonus as well) and ability to defend itself (As in: Cannot be attacked except by a Naval Unit at least 1 Era in advance of Phoenicia?) - and we've got the Trading, Seapower Phoenicia.