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The advantage being that the athletes get to stay with their fellow athletes helps them feel more at home
The advantage is that the athletes get to stay with their fellow athletes helps them feel more at home.
Are they equal?
Thanks.
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, ... the advantage being that the athletes [must be included in a larger sentence
The advantage is that the athletes
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I don't like either sentence; the second one is definitely ungrammatical. Being is most commonly used to indicate a reason in clause equivalents if it occupies initial position:
Being tired, I went to bed. (= Because I was tired, I went to bed.)
Being has other uses. In your example being doesn't have initial position, which is just fine, but there is a subordinate clause (that the athletes get to stay...) and a finite verb (helps) supposedly in a main clause after the advantage being. That jars in my ears. There is no proper subject for the finite verb. If the beginning of the sentence were used after a main clause, everything would be fine. For example:

The athletes are all housed in the same building, which everybody likes, the advantage being that they feel more at home.
Now there is no main clause after the advantage being... Other members may have other opinions.
In your second sentence you have a finite verb (helps), which has no subject. We could also say that there are two finite verbs (is, helps) in one clause (The advantage is helps them feel more at home). A relative pronoun must be added as the subject of a relative clause:
The advantage is that the athletes get to stay with their fellow athletes, which helps them feel more at home.

Cheers, CB