Why does this big button require 12V DC for an LED?
The LED and switch are separate.

The LED says it needs 12 V, this might be because they've connected several LEDs in series, or used a large resistor. No way to find out except to ask them, or buy one and look. If it is multiple LEDs you might be able to cut some tracks and put them in parallel, so they work at 5V

The switch is a 3 contact microswitch, so will be very easy to read from a small microcontroller. Your plan is probably the neatest and quickest way to get a switch to talk to USB.