So, it looks like the Project Server app is only going to work with a PostgreSQL 9.5.4 install.
It requests you run the Project Server installer again, and check the PostgreSQL 9.5.4 install option. After the Project Server install completed, launched it, and it doesn't see the PostgreSQL 14.1 install. On the installer, I unchecked the option to install PostgreSQL 9.5.4. I also installed the Resolve 17.4.1 Project Server app on the same PC. So as far as I can tell, PostgreSQL 14.1 works fine with Resolve 17.4.3. I then Restored the backup, to a new db called v17v2. The first time I ran a backup, it asked me to point to the PostgreSQL bin directory. I copied projects from the Disk database to the new v17 PostgreSQL db. I launched Resolve Studio 17.4.3 on the same machine, and created a new PostgreSQL database using all the PostgreSQL New Database defaults (Location 127.0.0.1, Username postgres, Password DaVinci) and gave it a name of v17.
I used the default Resolve postgres password of DaVinci when asked for the postgres pw. OK, I downloaded and installed the PostgreSQL 14.1 installer for Windows, and installed it on a Windows 11 PC.