printer to dock

Open System Settings : Printers & Scanners panel. Drag and drop your printer icon from the Printers listing to where you want it on the Dock.

I tested this on Sonoma 14.1.1 by dragging and dropping a PDF and a Word .docx onto the Dock printer icon and the latter document was converted to PDF and sent to the printer.

The downside of this printer functionality from the Dock is that the printer print queue panel remains on screen and needs manual dismissal, something that is not an issue when printing from an application and a dynamic Dock icon.

In /Utilities/Print Center, use the "Printer -> Add to Dock" menu item.

I have only seen this fail when the home directory is not on the system volume.

If that's not the case, then this is worth reporting to Apple. Sorry that didn't work.

Feedback - macOS - Apple