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Let's say the arduino did have more analog inputs. Say 32 or 64. You could connect all your inputs simultaneously, but think about your code: you've still got to read an input, do an action, go to the next input, and so on, so it's never truly instantaneous. You're going through a loop, and hitting every input along the way. Maybe the loop executes 100 times per second, so there's a .01 sec delay, at most, from your knobs. If it happens that you have to multiplex your inputs, this may add, say 1ms on top of whatever else is going on. It won't be instant, but for all intents and purposes, you'll likely not notice the time delay.
If you find that you need something truly simultaneous, the arduino probably will not do. Why? Because it works sequentially. It can look at an input, do something, and then do the next thing. A musical instrument, of course, can multitask: all the strings on a harp can vibrate independent of one another. If you find that you absolutely need this type of processing, the arduino may limit you. First, it is fast, but not breathtakingly so, and second, because it cannot process inputs in parallel.
There are two types of chips, called DSPs and FPGAs, which are more suited to high speed processing of many inputs. They are very different to program than the arduino, but they do find use in high-end audio applications.
That being said, I think the arduino, plus a multiplexer, will be more capable than you're inclined to believe, for sequencing MIDI. Multiplexers can work extremely quickly, and interface readily with the arduino.