That 1W is GREATLY more than the few milliwatts of your single transistor effort. Power output (with a correctly designed circuit) is dependent on power supply voltage, and speaker impedance. Using an output transformer, simply alters the effectvie impedance of the speaker.
I'm rather bemused it's called a 'Japanese radio', it's just a bog standard radio design back from the days of germanium PNP transistors (NPN either weren't available yet, or were even more horrendously expensive) - probably originating from a Mullard data sheet?. Once NPN germanium became reasonably priced, the transformers were mostly dumped, and quasi-complementary output stages introduced instead?.
I suppose it could be 'Japanese', as they started producing versions of the older designs, after European manufacturers had moved on to more modern transformer-less ones.