Interplanner,
You ask questions as though the answer were obvious and the implications of the answer that you have in mind were also obvious. But neither is the case.
First, what needed to be proven "right away"? The resurrection and messiahship of Jesus, and the reality and truth of the good news. The proofs of these things--which often but not always included signs and wonders--were not always successful, either among the Jews, or among the Gentiles. Even Jesus himself, who did signs and wonders on a regular basis, was not believed by all who encountered him and witnessed his works.
It's true that the Holy Spirit testified to the truth of the Gospel by doing works of healings and other dramatic and surprising things. But there is no biblical doctrine that says healings and prophecy are for the sole--and temporary--purpose of giving a kind of kick-start to the preaching of the Gospel. Wherever in the world people have not heard the Gospel before, the same situation applies as applied when Paul and others went out with the Gospel to people who had never heard it. The Holy Spirit desires, as always, to testify to the love of God and the truth of the Gospel.
And Paul talks in 1 Cor. 12 and 14 as though it is a normal aspect of Christian community worship and fellowship for the Holy Spirit to give out various gifts including prophecy and healing. It is a completely implausible reading of 1 Cor. 13, and an insult to people's intelligence, to suggest that when Paul speaks of the perfect coming, he's talking about the canon of Scripture. He's talking about the resurrection, not some fuzzy moment when all the writings of the NT have been committed to paper.
It is hazardous to hold, as doctrine, the idea that the Holy Spirit never gives out gifts of dramatic healings and prophecies and visions, just because it makes you nervous that someone might claim authority for false doctrine based on their own revelation or signs and wonders. Forming a doctrine that flies in the face of scripture itself, and that commits you to interpreting the gifts of the Holy Spirit as fraudulent, is a really bad idea.
That's not to say, by the way, that there are not a lot of bogus signs and wonders and prophecies out there. There surely are a lot of counterfeits. But it doesn't make you safe from error to throw a blanket interpretation of "fake" over every alleged manifestation of the Spirit. Things need to be discerned on a case-by-case basis with an open, unsentimental, un-cynical mind.