Electronic – Create an audio cable with 100k ohm resistors

Don't be so shy.

0.3W is pretty loud! And don't forget that audio equipment potentiometers are logarithmical ones, so when you turn it halfway, it is roughly one tenth of the output power. So a 20W amp, driven with its rated RMS input signal would output 2W RMS when turned up halfway. A quarter setting would result in 0.2W RMS accordingly. And you said you'd want to test with music - rated music power is around 2x the rated RMS power for most of the speakers.

Don't fret, speakers are quite difficult to destroy in so low power situations. I did try with a PC speaker and a ~20W amp when I was young and feeble, to the point when it got quite warm on the outside - but no pops (it was horrible, by the way)... Ok, PC speakers are not too flimsy. I did build a tube amp from scrap parts, and on that I regularly used a way too -electrically- underrated speaker, and to make the situation even worse, I used it as a guitar amplifier - driven into the maximal power tube saturation I could achieve...

If however you'd want a professional approach, use an LM386 power op-amp to play with, a few external components and you are ready to go. I think that is far easier to obtain than an impedance matcher transformator - not to mention, it degrades and colours the sound a lot less.

LM386 datasheet

Sounds like you should consider placing some good quality capacitors in series with the audio signal lines from the jack that you added and to where you connected the inputs to the guts of the old cassette player's amplifier input. I would start with a value of 10uF with a voltage rating of 16V or more. It would be best to utilize non-polarized capacitors but if you cannot find those then consider one of two options:

a) Use two 22uF capacitors in series wires + to + and then one - lead to the jack and the other - lead to the cassette input.

b) Try to see how a polarized cap would work. Put the + lead toward the cassette player amplifier and the - to the jack.

These capacitor ideas block the DC bias that is apparently present at the connection point where you attached the signal wires into the cassette amplifier board. Some of the earlier devices that you connected to the cassette jack may very well have had capacitor coupled outputs. On the other hand your newest device may not have an output configured like this.