The main "problem" is that that it's widely misunderstood by both those circuit designers that use it, and those that eschew it. It's also very much out of vogue these days. What negative feedback is, in essence, is the technique of trading circuit gain for circuit bandwidth and linearity.
But the vast majority of audio circuits use some form of negative feedback, regardless of whether or not they're advertised as "zero feedback". It's interesting to notice that many who shun feedback also prefer triodes . . . as most triode circuits have a good dose of negative feedback (based on the tube's internal characteristics). In fact, when a given circuit or active device (tube or transistor) displays the combination of less gain and improved linearity, it's likely there's some kind of negative feedback mechanism that's making it that way.
The common audiophile response is then "well that's LOCAL feedback, which is good!" . . . this usually is explained by lack of "delays" and such. But actually in many cases (i.e. typical solid-state amp), most of the "delay" (really a phase lag, NOT a pure delay) is in one stage (the Miller-compensated voltage amp), and most of the nonlinearity is in another (the output stage), so the stability consequences of global feedback are usually very similar to that of just local feedback around the voltage amp . . . but the global feedback arrangement of course works so much better. Incidentally, this is the main cause for higher-order/frequency distortion products in solid-state amps that use feedback - the distortion rises with frequency because the Miller-compensation technique shifts feedback from global to local as frequency increases, and takes the output stage out of the feedback loop. So the problem is really not with the feedback, but more the lack of it . . . that is, it's not available equally at all the frequencies that need it.
And it's also a misconception that local feedback results in automatically better stability . . . instability in cathode/emitter follower circuits from certain source impedances is a very well-documented condition. In fact, this is the whole reason for the invention of tetrodes and pentodes . . . triodes exhibit poor stability at the limits of their gain and bandwidth, as a result of their internal feedback mechanism. The addition of the screen-grid mostly eliminates the feedback, thus the increase in gain, and decrease in output impedance and linearity.
Oh and as for positive feedback . . . "bootstrapping" networks are extremely common in all kinds of analog circuitry . . . does that count?
