It sounds like the problem is not with the electrical contacts inside the switch (because the amplifier does turn on), but with the mechanical latch mechanism that implements push-on/push-off. And that's probably due to dried grease.
DeoxIT might actually work to clean the mechanical parts, but that's not what it's designed for (and it's pretty expensive). I'd clean the mechanical parts with something like naphtha (lighter fluid), then relubricate with grease after it had dried.
Dave