The 0.01W is presumably a maximum rating. If you could hear clicking during a DMM capacitance test then it's probably reasonably sensitive.
You need very little volume for a pillow speaker (that's kind of the point). I'd just try it and see if it is loud enough. From the screenshot, it just came with an ordinary phone (headphone) plug. Headphone outputs of the time didn't output much voltage if they were mainly designed for magnetic headphones.
EDIT. If you are wanting to experiment with connecting it to a normal stereo headphone output, you could try a series 33R resistor from each channel to one lead and return the other lead to the barrel connection. That will give enough resistance to provide a stereo to mono conversion and help isolate the capacitive load. If it works, fine. If not, then you'll have to decide whether it's worth going to more expensive solutions for greater volume.
Well I have good news and... other news. I made a cable with resistors as you suggested and the speaker works
Tip and ring conductors I kept together on the speaker side and the shield by itself. Unfortunately, I grabbed the wrong pack of resistors off the bench (I really need to do this stuff when I'm more alert and not at 5AM). I'd rather not peel layers of heat shrink, so it's staying this way unless it might break the speaker. Used 6k8 instead of the 33R you suggested. Not sure wtf that might do, but it works.
They're rated for 1/2 watt. With more resistance, I have to turn the gain higher which is probably safer for the speaker so it slowly gets louder. Or not, idk lol. At least it gets louder gradually.
To try and work out the polarity since I can't see the speaker cone, I didn't solder the new piece of cable together with the existing one, and touched the resistor leads to the side plugged in and listened. One way sounded more distorted with less bass so I figured that was out of phase and wired them the other way.
At any rate, it works. As one would expect it's heavy in the mids and high mids. You can make out the harmonics of bass and drums in certain music. Interesting sound. Gonna mic the speaker tomorrow, analyze it and mess with some plugins to shape the audio a bit. I like new toys.
Anyway, thanks for the help. I give myself a C-
P.S. soldering small resistor leads to audio cable conductors and thick shield is a pain in the ***