Two remaining reasons to give a speech: the handshake, and the hug

Before the advent of YouTube and before that email and before that TV and before that radio and before that the Gutenberg printing press, a speech was a pretty efficient means of communication. "Round everybody up, I've got something I wanna say!"

But that was a long time ago, and we are still giving, and listening to, a hell of a lot of speeches.

Why?

Mostly for symbolic reasons—and occasionally, for communication:

1. The speech as mass autograph-signing. In this time-honored social ritual, the One is relatively exalted, and the Many are relatively downtrodden, the gap between them such that each is flattered (but not gobsmacked) to meet the other, in benevolent or subordinate proximity. The message the One delivers doesn't matter nearly as much as his or her simply showing up, but a speech we must have, as pleasing aural filling between the ritual opening and closing applause. This is the chief purpose of most speeches, and the way most speechwriters earn their bread, most of the time: making ceremonial speeches sound and feel like communicative speeches.
2. The speech as a social communion. There are times when the One and the Many realize they truly need each other, to make their lives better, or to keep their lives from being ruined by a malign force. The Many need One to make them one. And the One needs to be made many. And both the One and the Many know what is needed: an open-air declaration of the intellectual and emotional truth of the moment. "We have nothing to fear but fear itself.""Ask not what your country can do for you, ask what you can do for your country." "Mr. Gorbachev, tear down this wall."  

This is the kind of work speechwriters dream to do, because it describes more than a useful job. In a secular society, it's the closest thing to a religious miracle. It's a human miracle, and you know it has occurred when the Many aren't gazing in admiration at the One, but in wonder, at one another.

Happily for professional speechwriters, technology won't replace either of the two remaining functions of live speeches any sooner than it will make obsolete handshakes and hugs.