I think in Korean, like how when we say ugly it means "was poorly formed", in the same way broken must also always be in past tense. When we talk about the broken tv, it is almost always implied that it is already broken. And 고장 났어요 is the past tense of the verb already. In English as well, 'broken' is in the past tense.
Technically, broken is what is called a past participle. A participle is a word has properties of both a verb and an adjective at the same time. Generally they are verbs being used as adjectives. The other kind of participle in English is the present participle. For most English verbs the form of the past tense and the past participle is the same, but for irregular verbs it's often different as in ate vs eaten.
The notes say it is a phrase consisting out of two words, so there is a space.
The word broken in English has pretty broad meaning, when you think of it. Something could be smashed, snapped, shattered, or cracked and be "broken." Korean doesn't have an all encompassing word quite like English. One of the more common ways to say something is broken is a phrase itself "고장이 나다."
고장 means something like "problem" or "malfunction" and 나다 means "arise" or "grow"
Based on this, it only has the meaning of something being broken so as not to work, not that something has been shattered or cracked in pieces.