Don't think you should focus on + 지 않다 (not/non-) when deciding to use 은(-ㄴ)/는. Look at the type of the main verb, whether it is an Action verb [AV] or Descriptive verb/adjectival verb [DV].
[AV]+ 은(-ㄴ) -> modified Adj, past tense [DV] + 은(-ㄴ) -> modified Adj, present tense
For [AV] to be turned into an adjective in the present tense then use +는.
[AV] + 는 -> modified Adj in present tense * 는 is not used for [DV].
Once the type of main verb and the appropriate marker/modifier are determined, + 지 않 will be placed between Verb and marker to negate the adjective/adjectival verbs.
In the example given, 마시 - drink is the main verb
(1) 마시다 is an AV
(2) To modify it in the present tense, +는 is selected
마시+ ... +는 = drinking/ who drinks.
(3) To negate the adjective, insert +지 않 between verb and modifier marker:
마시+(지 않) +는 -> 마시지 않는 = non drinking/who does not drink
And we have it.
(1) "who does not drink" is an adjectival clause. Its role is that of an adjective describing the noun which it is attached to; in this case, "the man".
"The man who does not drink" is known as a noun-phrase. It functions only either as subject or as object.
(2) Your suggestion: "The man does not drink" is an independent clause with a noun subject (the man) + verb. Unlike "who does not drink", "does not drink" (without who) does not describe the man. It describes his action.