The explanation you got is simply inaccurate. Indeed, ~なくて can express "because ~ not...", but it's not always so.
The main function of the -te form is to connect clauses mostly in the temporal order, and the meaning differs depending on the relation in meaning between the clauses. It might be just "and", it can indicate the cause/reason of the main clause, or even can be adversative conjunction "but".
The same goes to ~なくて.
e.g.
彼女は中国人ではなくて、韓国人です。
This なくて is not "because", and is interchangeable with なく.
The one who explained you might be confusing this なくて with "verb + なくて". "Verb + なくて" is often (in other words, not always) used to indicate the cause/reason.
e.g.
財布が見つからなくて、買い物ができなかった。
見つからなくて can be rephrased with 見つからなかったから/ので, and 見つからなく can't work here. Instead, 見つからず is used.
This is another example of the reliability of online resources. The answers you get from someone on the net (including my answers, of course) might not be always correct. You need to double-check it, as you did here. (You can show my answer to the person who explained なくて to you, and can ask what they think about it.)