Not so sure, may be improving bio availablilty of Fe in the solution

Yes it is possible, probably the reason that it is added in same concentration as Ferric ammonium citrate in growth media.

Does that implies that EDTA or other chelating compounds have no significant effect on Fe availability?

I don't think so, it might have to depend upon the ionic distribution, in certain cases and may chelate another predominant ion as well. 

In general citric acid maintain the ph of the media and simultaneously act as a energy source for the growing cell.