Your original construction exemplifies a perfect infinitive.You can also find it in constructions, not necessarily passive, like "He claims to have been busy", "He must have been busy" or "He ought to have been busy", all of which have a sense of indirectness or tentativeness about them. These constructions are not old-fashioned, but they would mostly likely be said by a careful and nuanced speaker. So you can capture that sense in those other two passive constructions 1 and 2, but note that only 2 uses a perfect aspect in the finite verb, whereas 1 is a simple past, so they are not equivalent. Also note that despite the past participle "said", your examples are all present passive. (If you find it difficult to unravel with the passive constructions, you can always recast them as active for the purpose of understanding: "Someone says that he was busy" and "Someone says that he has been busy".

Yes, "it is said" is a little formal and old-fashioned. You would usually find it in a fairy-tale setting or older literary text, or in a poem. The three verb constructions don't mean identical things. The first sentence is an unusual passive construction; first I'll analyze the original sentence: He is said to have been busy. (A continuous but definitely past action that is not going on now). This would be better expressed without the infinitive: People said he had been busy. (He is not busy now). 2nd sentence: also fixing the sentence to sound more modern, People said that he was busy. (An action that was occuring in the past but that might have continued - he might still be busy). 3rd sentence: People say that he has been busy. (This means that he has recently been busy and is still busy now and perhaps also in the future.) There are names for each of these verb tenses but I don't have my grammar book near me. As you can see they don't mean exactly the same thing.

Excellent question! I would consider all of these to be correct, but I would like to see the opinion of some others.