Is it possible to infer the characteristic curve of a recording device only from (many) recordings?
Yes, that's definitely possible. The key would be to find recordings where it's reasonably easy to find an acoustic reference. Great candidates for this would be classical or big band recordings. Let's say you have wax cylinder of Beethoven's 5th, you can try to create a reference by analyzing a dozen or so recent recordings of the same piece. The standard deviation would give you a "margin of error" for the reference and than you can simply divide the spectrum of the wax cylinder by the reference spectrum. Repeat for as many recordings as you can find good references for.


  should have an everage spectrum roughly resembling white noise


No. It's certainly not white: closer to pink but with more high end roll off. Hence you need a known good reference to determine exactly what it is.