The console toolchains are proprietary and locked behind NDAs. An engine cannot be both open source and support consoles. What it can (and Godot does) is have a closed-source version of the engine (or just plugins) that you are only permitted to license once Sony/Nintendo/MS has approved you. That version or those plugins can then contain the code required for the consoles.

Yes, exactly this with an added hurdle of Rust which isn't supported by any of the mainstream consoles.

An engine can be open source with proprietary extensions.