https://godotengine.org/asset-library/asset
There is in fact precisely such an asset library, part of the same database as the Godot startup menu (godotengine.org). The issue seems to be that when opening Godot itself, only the "Templates" "Projects" and "Demos" categories are visible: That website allows access to these but also other groups, like "2D / 3D Tools" or "Materials". Are those categories hidden by default on purpose or is this a bug?
[EDIT] I am a fool. The other categories do show up when looking at the Asset Library from an open game project. Sorry for being such a newbie, I shall look more deeply first before asking a question.