Regarding non-volatility. There is no simple way to add it - at least, not that I can think of. It's so obviously the kind of thing that a microcontroller would do that I don't even like to think of how to add it to a "dumb" design.
I think the answer would be to have the 40193 powered from a battery-backed-up supply rail. This requires that any connections between the 40193 and the parts of the circuit that are not powered during backup must behave in such a way that, while the 40193 is powered up and the other parts are not, (a) voltages from the 40193 to the rest of the circuit do not cause damage to the other ICs (ICs are not supposed to have voltages applied to their inputs when they are not powered up); (b) other parts of the circuit do not draw current from the 40193's outputs (since this current would come from the backup battery and would cause it to discharge); and (c) that the signals from the rest of the circuitry to the 40193 remain in known states so the 40193 doesn't accidentally get reset, or incremented or decremented, or loaded.
Actually I think we could get away with just one extra IC for non-volatility, but I'm not sure. Have a think about it, and think about these next comments, then let me know whether you want me to design a non-volatile circuit. You might also want to consider how important non-volatility would be, i.e. how often you would benefit from it (how often you would turn the unit OFF and back ON and want the input setting to be preserved) and how much extra work it would take to just press a button up to 8 times to re-select the input. And whether you want to add a lithium coin cell which will need to be replaced after 10~20 years and could possibly leak.
I agree that a row of LEDs would be better than a two-digit 7-segment display.
I agree that having all of the LEDs flash when MIDI data is detected on the respective input would be a very good idea. This can be done in various ways. The simplest way would require three extra small components per LED: a resistor, a transistor, and an electrolytic capacitor. That simple arrangement would give a glow that fades out, rather than switching OFF cleanly. If you want a clean turn-off, add another few small components per LED, or a few more CMOS ICs.
You can get three-wire LEDs that contain a green LED and a red LED (and other combinations are available) in a single package. You could use one colour (e.g. green) to indicate "selected" and the other colour (e.g. red) to indicate "activity". For that example, the selected input would indicate green, or orange (green + red) while there was data, and unselected inputs would be unlit while idle, and red while idle. As I said, you can get other colour combinations, and we could also arrange things so when there's data on the selected input, the LED changes to the other colour only, instead of showing both colours, if that fits better with what you want.
Another option would be a single-colour LED for each input, as you suggest, and combine the selection and data signals so that the selected input is steadily illuminated, and the unselected inputs are steadily OFF, but data (in both cases) toggles the LED to the opposite state for a short time. So unselected inputs would blink ON in response to data, and the selected input would stay ON and blink OFF in response to data. This would be confusing to look at though, I think, if many inputs had data on them simultaneously. That's why I thought that bi-colour LEDs would be better.
All of this control would be much more easily done by a microcontroller, but as you say, that's quite a jump in complexity (not hardware complexity, but overall difficulty).