The LM386 amplifier's inputs are an internally biased differential pair, with common mode input range extending to the indicated ground, Vee, or actually 0.4 Volts below ground.
In order to not disturb this internal biasing, the input signal must be capacitively coupled (or perfectly ground-referenced) - a capacitor in series on the input line is needed.
Next, for a voltage gain of 200, the input signal peak to peak needs to be under 1/200 of the output voltage range of the amp. While I could not find this specified in a quick scan of the datasheet, if we assume 7 volts output voltage range with a 9 Volt supply, the input signal needs to be under 35 mV peak to peak, to avoid clipping the signal. Clipping would result in mild to severe distortion of the output - though this does not explain the complete lack of signal you find on output.
If the incoming signal is higher than this 35 mV P-P, a potentiometer as an attenuator on the input is suggested.
Once these fixes are done, please revert with results, so the answer can be added to if needed.
It's not going through you.
According to your drawing, the reason this works is actually the opposite of what you intuit. The audio signal isn't going through your body. It's going through the wire.
The output from the PC audio port is AC-coupled which means that it is pushing and pulling on the mobile charges in the wire with an electric field, not direct conduction. More simply, there is a gap between the wire touching the PC's internal amplifier output and the pin of the audio jack where your wire is connected.
In order for the amplifier to receive this pushing and pulling on the charges (the audio signal), everyone (PC and Speaker's amplifier's input) has to agree on what zero means (e.g. no pushing or pulling). In physical terms, this is the point where the speaker's diaphragm is in the center (neither pushing or pulling on the air) -- where it sits without power.
To provide this "zero" (we call that the reference potential or just "reference"), the second wire is included in the cable.
When you only connect the signal wire and not the reference, the amplifier receiving it doesn't actually perceive that the signal is being pushed or pulled on because the entire amplifier moves up and down. It's like being on a perfectly smooth and flat surface while standing on a skateboard and having someone pull you on a rope. All of you translates.
But if someone takes you off the skateboard and then pulls on the rope, your feet are stuck and you fall forward (you feel the rope pulling on you relative to your feet).
In your audio case, your body contains a giant mass of mobile charges. When you do not touch the reference contact of the audio connector (the area at the back of the TRS plug closest to the plastic overmoulding) it is in contact with air which has almost no mobile charges and it's like standing on the skateboard. When you touch it, you add (electrically) your entire body worth of mobile charges. This is a stable enough reference to allow the amplifier to see the wire moving as distinct from you moving.
This effect is the basis of Capacitive Touch Screens.
My explanation is an accessible approach to understanding how capacitance in general works. A similar situation to the one you described explains how capacitive touch screens work (as are found on modern smart-phones, like the iPhone).
The screen is transmitting the equivalent of an audio signal all the time and you touching the screen adds your body's mobile charges to the picture and distorts the screen's "audio" signal. It is this distortion introduced by your finger that is detected, not your "touching" of the screen. That's why capacitive touch panels don't work well (or at all) with gloves or when your fingers are wet. These conditions change the way your finger distorts the electric field and confuses the detector.
Best Answer
Modern analyses of electrolytic capacitors definitely show an increase in ESR and a reduction of capacitance but these are modern capacitors and your last-century caps might be worse. See this document entitled "Experimental Studies of Ageing in Electrolytic Capacitors" and particular the graphs on page 4 and 5.
DC leakage currents may also increase with age and this could alter the bias points on the amplifier for your electrostatic speakers which, in turn, might cause higher frequencies to be attenuated more hence, your perception of better top-end response when the caps were swapped. See also this document entitled "Capacitors Age and Capacitors Have an End of Life".
If just used as a high-pass filter I expect that you might not have witnessed any aural change but given that bias points on amplifiers could change I think it's feasible.