That's the theory, but we're mimicking a UART here so the hardware doesn't do it. The handshaking signal can only cause an interrupt and then it's up to software to stop sending. And it's up to software to notice the buffer's getting full and set the handshaking signal in the first place.
But the maximum rate that most UARTs can send is a 115.2Kbps. Even the tiniest modern microcontroller can receive that rate without overflow, assuming interrupts aren't disabled too long, and if they are then handshaking wouldn't work anyway. And we're only doing 19.2Kbps here.
The real reason they used handshaking was that an extra few bytes of RAM for a buffer was very expensive, so they absolutely had to process (e.g. print on the teletype) all data as it was received.