Incidentally, the TRS-80 Model 100 portable bit-banged cassette and telephone modem from two serial pins on the 80C85 itself. If memory serves, the "transmitter" was simply a low-pass filter.
Here's a KCS implementation from Steve Ciarcia's book, BuildYour Own Z80 Computer. The author warns that it takes a lot of tweaking to get a higher rate than 300bps from this circuit.
If it were me, I'd build the interface from a Raspberry Pi Pico or similar.