Translation:The man who does not wear a tie cannot speak English.
This is a much better translation. The current one is actually about a man who habitually doesn't wear ties, which doesn't make sense as a way for the speaker to indicate what man they mean. How would the listener know that guy's habits? Both speaker and listener can see his current state of dress though.
The whole Korean course so far uses the English present indicative where every native English speaker would use the present continuous. The unqualified present indicative is mostly used in English to talk about customary habits and generalizations. The course believes we only use it when Korean also its continuous. Clearly the two languages work differently and pretending otherwise is not helpful.
Now perhaps they only do this in the early part of the course to keep it simple and then go on to use them correctly later in the course after we've learned more. I would accept that I suppose but in the meantime we can expect this kind of strangeness.
Do you mean the Korean sentence is awful or the English translation? Please give us better ones. People are still trying to correct "All that glitters is not gold", and proverbial "A stitch in time saves nine" etc. Perhaps you can help convert them to non-awful sentences also.
(1) 매지 않다 = 매(다) + 지 않다 (2 separate parts)
매다 = to tie, fasten by means of a string or something similar, to knot
-지 않다 = do/does not [long negation pattern]
(2) 넥타이를 매다 = to tie a tie - standard usage. (Also, used with 신발끈 = shoelaces)
Other synonymous expressions:
넥타이를 하다 -> 하다 = to do. Used when the correct word does not come to mind.
넥타이를 입다 -> 입다 = to wear. Should only be used for items covering your body.
넥타이를 묶다 -> 묶다 = to strap; to secure with a string or something similar -> best synonym for 매다 although 매다 focuses more on the action (= to make a knot) while 묶다, more on the aim of the action (= to tie in a secure manner).
In English, the verb “to wear” is used for all clothing items but in Korean depends on the item. This website explains very well all the verb possibilities that exist in Korean: https://wiseinit.com/입다-신다-쓰다-끼다-차다-두르다-메다-매다-바르다-붙/