V1て + V2 usually means "V1 and (then) V2", in which case V1 and V2 are just two actions placed in parallel. However, you may know
V1て + V2 sometimes means "does V1 before V2" or "does V2 with/while/by/after V1". In such cases, V1 is essentially an adverbial modifier that expresses how V2 happens.
I eat meals using chopsticks.
I listened to him silently.
What happens when a sentence like this is negated?
I don't eat meals using chopsticks.
(I do eat meals, but without using chopsticks.)
I did not listen to him silently.
(I did listen to him, but not silently.)
You can see "not/ない/ません" is negating the adverb part ("using chopsticks", "silently"), not the main verb itself! Here, "eat" and "listen" happen anyway even though they are (directly) preceded by "not".
So this is what is happening in your example, too. ない is negating the adverbial V1 part, not the main verb (V2), even though ない is placed directly after V2.
When V1 is adverbial depends purely on the context. In the following sentence, V1 is not adverbially modifying V2, so ません just negates the verb before it, V2:
He drinks only water and doesn't eat meals.
You have to get used to this "adverbial V1" pattern through many examples. Here are more examples of adverbial V1:
I studied before taking the exam. / I took the exam after studying.
I did not study before taking the exam. / I took the exam without studying.
students who walk to school
students who don't walk to school
(Not "students who walk and don't go to school". Compare: コンビニで働いていて学校に行かない学生 "students who work in a convenience store and don't go to school")
He drank and drove, but she did not drink and drive.
(She drove, but not under the influence of alcohol.)
I did not brush my teeth before going to bed. / I went to bed without brushing my teeth.
I tried on the skirt before buying it. / I bought the skirt after trying on it.
I did not try on the skirt before buying it. / I bought the skirt without trying on it.
(This is technically ambiguous; it can also mean "I tried on the skirt, but did not buy it" depending on the context.)