We do. Every American port. Port authorities lease operating spots to private industry. Importers and exporters may own property near port operations but the ports themselves are usually owned by local government entities.
.....operating spots .....
You're on the right track but to sharpen it up, they lease space for terminals. The flap in question is about P&O terminals operating in the various ports.
The key to efficient steam ship operation is efficient terminal operation. A ship is in a different American port every day or so as it proceeds on it's tightly scheduled voyage. First New York then Baltimore or Norfolk then Charleston or Savannah. Then back across the Atlantic. To schedule a regular weekly or biweekly call, it cannot tarry in the terminal. The terminal must act speedily to load and unload the vessel. The terminal that does the best job lowers costs for the steamship line in a very competitive business.
The Dubai company that is by the way owned by Arabs but certainly staffed by American and Brit and Indian managers has sharpened and honed these logistic skills to first class sharpness.
The hue and cry by the anti's is total blather based on extreme ignorance.