What does this extension do?
I played a lot of RA2:YR in my youth but am not in the RA2 modding scene and I couldn't really understand how to use this extension or what it might provide.
Basically it patches bugs found in the original game and provides new engine features. RA2YR engine is not like modern game engines, you don't have a scripting language, everything is configured via config files without the need to code, hence modders can't implement custom game mechanics apart from what the engine provides. For example, per-unit shields with additional HP and different armor were not possible to do, with Phobos it is possible.
Without reading the whole page (there is a lot), do I get benefits from installing this on a bare installation?
I want to keep it vanilla, but I want the bugfixes like NightHawks & Rocketeers not being able to chase & fire.
The bugfixes you said are not really something fixed by Phobos, IIRC those can be fixed using INI editing.
Without modifying the game some occassional bugfixes are applied anyways. Among those are elevation overshooting bugfix (not very visible ingame because of artillery not using arcing projectiles in vanilla), some units "dying" multiple times because of damage applied to them during dying, maybe some else.
This is like 2% of Phobos though. As said, this is mostly a tool for modding, so to use most features you need to modify the game's INI files.
Electronic Arts sitting on franchises like this... by the dozens.
Is there a video which showcases the unique aspects of RA2 gameplay with Phobos?
Scroll through the extension docs:
https://phobos.readthedocs.io/en/latest/New-or-Enhanced-Logi...
They have recordings of most of the different features Phobos enables.
Hopefully there will be in future. But for now the docs do have fancy GIF illustrations of Phobos features used, which is the closest you can get.