Because semantically it requires a different syntactic construction to represent this.
This lesson is about modifiers, ergo, the key modifier here is the adjective "running". It wants to show you how to adjectivalize a gerund specifically. Further, this is in the present simple tense. What your sentence accomplishes, while still more or less similar, is a demonstration of the present progressive tense, which by now we have not learned yet.
One way of understanding this is that you mixed up what is new and old information. In "the fat man is running," it's already understood that there is some fat man, and you're telling the listener that he is running. In "the running man is fat," it's the other way around.
Of course, context helps in cases like this, but those two sentences definitely do convey different meanings.
는 is not added to a descriptive verb (adjective)--뚱뚱한. Also, Koreans often express the continuous tense with the simple present.
The key to this sentence is what pluckingstrings has pointed out. The focus of the original sentence is on a man running, who happens to be fat. You could change fat to any of a number of descriptions-- The man running is slow. The man running is tired. The man running is healthy. The man running is ugly. etc.
"The fat man is running." focuses on a fat man who happens to be running, which you could change to a number of activiities. The fat man is swimming. The fat man is eating. The fat man is belching. The fat man is floating. etc.