So first off, I don't think you need 3 axes.
You can easily do this with two axes.
You got 3 "positions": input CD pile, CD reader(s), output CD pile.
That's something you should be able to manage easily with a 2 axis setup (horizontal+vertical).
Plus a relay/SSR for a simple vacuum control setup.
I'm not sure if I'd go with a corexy system, it's more difficult to get right than a straightforward axis-on-axis setup if you're a beginner (you have to be very serious about tightening and tolerances and wrapping etc), and I'm not sure if you actually need the extra speed/light weight.
If you really need speed here, a great option would be a flattened/2D version of the linear delta architecture (or if you really need 3D movement, an actual/non-flattened linear delta structure). The 2D version of linear delta is much simpler to get right, it's just two linear axes, a few ball bearings and 4 long "plates".
I wouldn't worry too much about the learning curve, nowadays, there is a ton of great documentation online, and you can use chatgpt for guidance/advice, and there are great communities like this one too.
For control software/firmware/hardware, I'm the guy that created the smoothieware project so probably won't get very objective/un-biased advice from me about that, but what you'll get is help if you ever need it.
In general, I think if you just go with a simple gcode-based control board (like openpnp uses), and a simple xy setup, it should be fairly straightforward to get something going that works well.