Analog Potentiometer Output to USB Serial
A good place to look at are the Microchip microcontroller. You can go from the PIC18 to PIC25 or PIC32 range depending how many pots you need.

The PIC32MK can have up to 42, 12-bit ADC input, so you could connect 42 pots directly on a single 5-10$ chip.

Then they natively supports USB and Microchip have what they call the Harmony stack which makes it "simple" to configure the USB through a (bit buggy) visual configuration that generates the code for you.

Microchip provides the stack, have cheap programmer and are fairly easy to use, but you will find similar MCU in a lot of other manufacturer.

It's never easy to start from scratch though and be ready to scratch your head quite a bit to get it working, but once you pass the first wall it becomes very valuable knowledge.

You can use the USB Device mode, which is better than serial because you can directly bind the computer side software to the usb PID and VID, and not mess around with COM port settings that changes all the time.

I do not think USB supports natively MIDI format, so you would need to have a computer side software that takes the data from the USB device and transform them into MIDI and forward it where needed.