as a native english speak, i agree. This course is showing me how much less...clunky...other germanic language are. Seems almost like english take a round-about way to say everything while the other, less french-influenced Germanic languages just get right to the point.
furniture is what's known as an 'uncountable' noun. In English, it's impossible to have 'a furniture', just like it's impossible to have 'an ice' or 'a money'. You need to specify a singular amount. So 'A chunk of ice' or 'a fistful of money'. In the same way, you need to specify 'a piece of furniture'.
I think you explained it well.
The problem with the initial sentence is the word 'et'. This means the sentence is an indefinite singular such that the bed is AN important something. In this case the something is 'furniture'. As you, and others, have pointed out furniture is uncountable in English and needs to be qualified to become countable, and it is qualified by 'piece of'.
This is why 'The bed is an important piece of furniture' is correct, while 'The bed is an important furniture' is not.