The show is from the '60s, so why are they using an old wind-up Victrola???

For the same reason Sam Drucker uses an old-fashioned "candlestick" style telephone on PJ and Green Acres. These types of out-of-date props reinforce the idea that Hooterville operates at a much slower pace than the whiz-bang modern world of the 60s out there, backward you might argue.

The Shady Rest's wall phone was not even hooked up, further showing there hadn't been modernization. Along with an elevator that isn't one. The Cannonball itself is an old-fashioned steam train while the rest of the C&FW line consists of modern diesel locomotives. You can argue it's an unfair or unrealistic stereotype to show rural (and especially southern) regions as not having caught up with modern living -- but these anachronisms sometimes contribute to the humor, while providing a nostalgic charm.