There is really only one way to have silent switching. That's to do the muting part of the signal that's being switched out and the enabling part of the signal that's switched in by fading out/in over a few milliseconds.
Well, OK, two ways. The other way is be be very lucky and switch both coming and going signals at a zero crossing. Even then, the human ear will insert a discontinuity.
Any time you suddenly change a signal, the ear perceives a "tick" of varying degrees. Metal switches switch **instantly** from conducting to not and vice versa, and that's hard to keep quiet. Especially when they bounce.
In addition to switching transients, you can make it worse by changing the DC level of the switch instantly, as with leaking caps, different DC levels and so on.
Also some more subtle ways. Audio circuits vary in how susceptible they are to noise on their power supply (or supplies) and on ground. In the audio biz, power supply rejection ratio (PSRR) is a term you hear quoted a lot; that's the amount that any noise on the power supply is attenuated before getting into the output. Good audio stuff has PSRRs of 40-100 db, bigger being better. Simple single-ended transistor circuits may have PSRRs of 3db. Worse, they may *amplify* power supply noise if it gets into their bias chains. That's why bias strings are decoupled with caps - to try to keep power supply noise out of what is, after all, another signal to be amplified.
Ground noise is also pernicious. Many circuits have 0db rejection of ground noise. Diffamps tend to be rated in terms of common mode noise rejection ratios (CMR). Really good differential or transformer coupled circuits have CMRs of 60-120db (!).
Power supply noise and ground noise is what the "LED tick" is doing. The LED current is suddenly changing either the power supply or ground voltage at the amplifier's sensitive points by suddenly changing. It's usually a wiring or wire routing problem.
Any sudden change in current in a current loop also broadcasts itself as either a magnetic pulse to be picked up by another current loop, or a radio pulse. This is why wires are twisted - it decreases the area of the current loop, which makes the loop less able to transmit and also less able to pick up other pulses.