"Sétálni" actually means "to stroll" (to take a walk) in English. "To walk" is simply "to move on foot", it does not imply the speed, aim or determination.
"Sétálni" is to walk slowly, taking in the scenery, enjoying the fresh air. There is no generally valid translation for "to walk". It could be any one of:
menni - to go - without specifying the means
sétálni - to walk slowly, to stroll
gyalogolni - to go on foot - as a sport, or when emphasizing the mode of transportation, as in "let's not take the car, let's walk". And it has another version: "gyalog menni".
So, please keep this in mind. In real life, "menni" would be the closest translation of "to walk".
As you pointed out, "menni" doesn't specify any means of travelling. This is a huge difference compared to "to walk". "Menni" has the emphasis on the fact you are in the process of getting from one place to another, it's rarely considered an activity on its own. "To walk" has the emphasis you use your feet to move, no matter whether you are going somewhere specifically or just hanging around.
(By the way, according to Cambridge Dictionary, the most obvious meaning of "to walk" is, exactly, "to move along by putting one foot in front of the other, allowing each foot to touch the ground before lifting the next". That is, "sétál". Not "fut" or something.)
"Munkás" is mostly used for physical laborers. Not even necessarily employees. "Employee" is a much wider category. The best translation for "employee" is "alkalmazott", and sometimes "dolgozó".
So I guess "munkás"is used in a narrower sense than "worker" is. Maybe "worker" and "dolgozó" are closer in their usage. Maybe this is the best pairing of these words:
laborer - munkás
worker - dolgozó
employee - alkalmazott
But this is not set in stone, there is much overlap.