I'm not too familiar with the handlink, but the later seasons' version just has some LEDs that flash in different patterns if I'm not mistaken. A microprocessor would carry out that function very well, especially if you had a seven segment/matrix display IC in the design. The MCU would be able to handle the different levels of function pretty well.
Now, as far as practicality is concerned, if you can get a Pi Zero programmed and working easier than you could an Arduino, there's really no reason not to use a Pi. The cost is also negligible since those Pi Zeros are so cheap. The Zero W is $10 with some availability issues. In fact, if Wifi/Bluetooth wasn't required, the Zero boards sell for only $5, which would actually be more cost effective than using a microprocessor with a display driver, plus all of the extra bits.
Making a PCB with a socket for the Pi would be really cool to see too.
Any plans on making the first version of the handlink?