The race seems to be broadcast over the radio - how come?

Its not a radio broadcast. It is exactly what you see, a guy in the back room of an illegal gambling den reading the real-time Western Union tickertape race results on an intercom to the bookies and patrons on the floor. Radio broadcasting of races was illegal in many places because the tracks and local governments wanted bodies in the seats and to discourage off-track betting they couldnt control. So Gandolf was running an illegal gambling den that was unsanctioned which is why it was a basement facility with no sign outside. Sanctioned betting required expensive liscenses and were closely controlled by the tracks and local governments, just like casinos are today. But real-time Western Union results could read by anyone who owned a teletype, a phone connection, and reason to get the results, like newspaper reporters. Thats why the (fake) FBI raided the place - illegal off-track betting. Betting houses hired backroom readers because you can't have a bunch of people crowding the only teletype in the place. So, even if it sounds like radio its not. Every patron in the place knew a specialst in the backroom was simply reading the WU race results as they came in, but as this con showed, the local WU office was delaying the results just by a few minutes.

Yes, in the movie it's a fake broadcast being done by a guy in the back room. However, broadcasts were delayed by various means at that time. They were often recorded on disc for repeat in other time zones or at a later date, or even recreated as a performance. My dad was a radio announcer and disc jockey in California, and one of the things he remembered doing was getting a script on the wire from a ball game that had been played in the east, then reenacting it, with crowd sound effects, as if he was announcing the game live. They would even have a sound effect for the bat hitting the ball.