I think you meant to write that Fabrizzio was very anxious to go to America, Pherdy.
Sure he was anxious! But, apparently he couldn't get to America without help from the Barzini Family. Or, perhaps, they promised him something (like his pizza parlor) if he waited until they were ready to bring him over.
This leads to another speculation: We know for a fact that Fabrizzio didn't get to America until 1956, because Neri says so. But he killed Apollonia years earlier--perhaps as many as 10 years earlier, if you believe (as I do) that Sonny's and Apollonia's murders occurred in 1946, or 1947 at the latest. And even if you believe that Sonny wasn't killed until 1951 (as some here do), that'd still leave five years for Fabrizzio to be on the lam.
How did he survive? One possibility: when Vito said, after learning about Sonny's murder, "I want no inquiries made...this war stops now," he gave a reprieve to Barzini and Tattaglia that may have extended to Michael's need for vengeance for Apollonia. We might assume that Michael and Don Tomassino would be the ones to try to hunt down Fabrizzio, but if the Don gave orders from America, they'd have to obey. It's also possible that Fabrizzio fled Sicily for other locations where Mafiosi on the lam have traditionally holed up (the mainland of Italy? Tunisia? Gibralter? Turkey?), and lived there on handouts from the Barzinis until they were ready to send him to America. They'd have an incentive to keep him on ice until after the Don died because, if he were captured, he might be forced to blab about Barzini ordering him to blow up Michael's Alfa Romeo.