Jim, thanks for those references.
I think you've, by implication, summed it up. For my family, in the Shenandoah Valley of Virginia, the phrase 'old-timey' was a descriptor that could be applied to almost anything. It meant, and means, an object or action from another, older, time, with the occasional addition of implying that it is quintessentially that. Examples of this would be an 'old-timey' clock, an 'old-timey' recipe for pie, or an 'old-timey' wood stove. The 'old-timey' Hamiliton Railroad Special pocket watch was 'the' railroad watch.
I don't dispute that some folks may have used Old Timey in place of Old Time as a musical genre, but 'Old-Timey' for us has always been a general adjective.
Less often, and not by Valley folks, have I heard someone speak of an 'Old Time' car, clock, but I have heard it.