To transmit one's voice over the radio, you necessarily need to use a microphone, likewise with recording your voice. Microphones are just a way to turn sound into electrical signals, nothing more.

In terms of the first "world leader to use a PA", the particular honour goes to Woodrow Wilson in 1919. On the 9th of September he addressed fifty thousand in San Diego using Magnavox's wonderfully named "Sound Magnifying Phonograph". The microphone was a carbon granule type, basically a crude condenser microphone, which is rather low quality and has been superceded by better technologies. However, the speakers themselves were basically the same design that we use nowadays!