Did the American Italian Mafia actually offer "protection" to those from whom it collected "protection money"?

While you’re waiting for an answer to your question specifically, let me highlight a few different previous answers that’s are sort of parallel to your question.

You may first be interested in /u/BaronBifford’s answer to:

In the beginning of the Godfather, Don Corleone is seen doing favors to a community of Italian immigrants in New York, did the mafia in America actually do this during this period or was it just a romanticized view of post-war organized crime?

He summarizes Diego Gambetta’s point about the Sicilian mafia: that their primary purpose was as a third party that guaranteed things that the normal government couldn’t guarantee. /u/BaronBifford summarizes an example from Gambetta’s first chapter, which really is his main point about the mafia more generally (I happen to be reading the book right now):

This an illustration that Gambetta provides in his book of Mafia protection in action: Say a butcher wants to sell some meat to a supermarket, tax-free. Because this is essentially a black market deal, neither the butcher nor the grocer can sue the other in court in the case of foul play. What happens if the butcher sell some bad meat? What happens if the grocer doesn't pay the butcher? Each party is tempted to cheat the other and the mutual distrust may prevent them from making a profitable transaction. To prevent cheating, the two parties will invite the local mafioso to oversee the deal. The mafioso promises he will punish either of them if they cheat the other (by a sound beating, I should think). Because the mafioso is so feared for his violence, self-discipline, and uncompromising nature, neither party dares cheat the other and the transaction occurs smoothly. In exchange for his oversight, the mafioso takes a cut of the sale. Or maybe he'll ask for some favors in the future (and his clients better please him if they know what's good for them).

The Sicilian mafia did start out explicitly providing physical protection, especially to the landowners of Sicily’s vast citrus groves. Many mafiosi, Gambetta mentions, of course weren’t above committing acts of violence to drum up business for their protection services and committing against of violence against those who weren’t willing to pay, but with the weakness of the state, many landlowners (and the renters who actually ran the citrus groves) weren’t actually able to rely on the state to protect their property—which is why the began turning to mafiosi. While in America, we associate the mafia with dense urban areas, in Sicily, perhaps as late as World War II, the mafia were primarily (but by no means exclusively) a rural phenomenon, concentrated mostly in the parts of the island with industrial agriculture. Once established in providing protection for the bigger estates, the mafiosi could begin to sell protection more generally, such as to our butcher and grocer above.

While he doesn’t go into the American mafia, so I will have to leave that to others, Gambetta emphasizes that the main thing that Sicilian mafia deal in is not violence, but trust, information, and above all protection. They function as a quasi-, entrepreneurial state in areas where the state is weak, dysfunctional, or non-existent, he argues.

/u/BaronBifford argues that this likely holds just as true for Italian-immigrant heavy areas of New York, where the American State was not an effective means of resolving conflicts (BaronBifford reminds us that the movie the Godfather begins with an honest citizen making a plea to the titular Godfather because he, the honest citizen, was unable to get satisfaction through the courts), though he admits he isn’t an expert on the topic.

(I put the mafia in context with other non-state groups on the periphery of empires here).

/u/mikedash (author of a book on the American mafia and one of my two or three favorite AskHistorians writers, so the person I’d most want to read an answer to the question from) hints at the origin of the mafia here:

In the Godfather part II, the mafia emerged only after a proto-organized crime group, the Black Hand, was superseded. Was there anything that distinguished the mafia from this group, other than who was giving orders?

Before we had the mafia offering “protection services” (usually to a clearly defined “turf”), we had more straight up extortionists of so-called independent groups Black Hand which were more openly (and flamboyantly) threatening violence in exchange for a one-off payment, rather than the more regular and smaller payments for consistent “protection services”

I should also point out that Mike Dash gives a more historical (as opposed to abstract sociological) account of the rise of the Sicilian mafia in his answer to this question:

Why were mafia families mostly from Southern Italy

Mike Dash also has an interesting answer to the question of why there was never a German mafia in the United States, which mainly seems to answer that the German for various reasons didn’t end up in concentrated ethnic slums where the police and state had little effective presence and they also weren’t denied opportunities because of their ethnicity in the same way as other immigrant ethnic groups. These slums proved the breeding grounds for American organized crime, it seems, in that they left a vacuum for other groups to come in and challenge the state’s “monopoly on legitimate violence”—the same vacuum that had always allowed these groups to thrive.

These I think collectively hint at answers to your question, though I hope someone can provide a more definitive answer rooted in the actual history of the mafia in America.