It is possible to form a club that will be run by only one
person, in other words all the members yield the
responsibilities of buying and selling to one individual.
Can this individual that will run the club gain shares for
that service ?

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I have never seen this done, but I have thought about this situation before.  In my opinion this 'club' would actually be a mini-mutual fund or hedge fund and thus, the person in charge might have to be licensed. If the membership were limited - to perhaps 25 people or fewer - then I doubt that a license would be required.  Keep in mind that what seems obvious to you or me might be contrary to our laws. and it would be prudent for you get this information from the SEC or an attorney.

As to compensation, the membership could agree to pay anything it wants.  Funds usually pay their managers in cash, but there is no reason why the manager could not be paid in shares of the 'club'.