No, that isn’t a “commons”. And a commons in terms of the “tragedy of commons” isn’t well defined either. And the “tragedy of the commons” isn’t well supported either.

For example, sunlight. Sunlight delivers huge value to society but profit to no one. According to your claims that is a “commons”. And according to your theory unless a government manages sunlight it will always result in a “tragedy of commons”.

But we can even extend this to human created stuff. Especially scientific and mathematical knowledge. Lots of knowledge has been created that benefits everyone but does not deliver profit to anyone. For example calculating the area of a rectangle or a triangle.

These things still fit your definition of a “commons” being things that “deliver huge value to society but no profit to anyone”.