Good afternoon and thank you for posting on the Final Fantasy XIV Technical Support Forums.

It looks like you have done a lot of the things we normally suggest, however, we do have a few more troubleshooting suggestions.

Please make sure that "Cookies" and "JavaScript" are ENABLED on the PSN Browser. This can be done by going to the web browser on the PS4, press the options button, then go to settings. The options for Cookies and Javascript will most likely be checked, uncheck them and then recheck them.

The fault may also lie with the "Date & Time" settings on your PlayStation 4. We would advise to please follow the provided steps to adjust your console's clock to assure it is accurate and up-to-date:
Close the FINAL FANTASY XIV game/application.
Go to [SETTINGS] on your PS4.
Select [Date and Time], and "Set via Internet". (Please do not manually set the date and time.)
Once, the date and time settings have been updated, please restart your PlayStation 4 console.
Once your console has fully restarted, please start the FINAL FANTASY XIV game/application.
Please make sure all other internet-enabled devices (including smartphones, laptop/desktop PCs, tablets, etc.) have been disabled and/or turned OFF to minimize game interference.

Please let us know if you are still having trouble.

We hope we can be of assistance in helping you enjoy the exciting world of Final Fantasy XIV!