According to this, the popping is a symptom of a failed system check on boot: https://repair.wiki/w/New_Nintendo_2DS_X... It states there are 4 components that must be validated: Upper LCD, lower LCD, inner camera, and battery. If any are damaged (or their ribbon cables, or ribbon cable connectors) it produces that popping sound. Note that the ribbon cable connectors for the lower LCD power and RF antenna are NOT ZIF connectors like the rest. Lifting the "latch" on these breaks them such that they do no longer make a good connection. This is a common issue. "Five ribbons must be connected to the motherboard to get it to boot to the home menu. If any of these are not connected or are damaged in a way that the system detects as incorrect the blue light will turn off and the console will not boot. These five are the Lower LCD, Lower LCD backlight, Upper LCD, Inner Camera, and the Battery (see Fig. 3). Only these five need to be connected for the system to start."
This usually occurs when you break the lower screen's power ribbon cable connector. It is not a ZIF connector and "unlatching" it will break it causing a loose fit. The shoulder button presses against the ribbon cable causing it to shift in the connector resulting in the speakers popping and the device turning off.
Not ideal, but you can use replacements for DSi and (not-new) 2DS which are very common. This worked for me. They seemed identical except for 2 features:
1. New 2DS has a single centered leg in the front. DSi has two in the front on the sides.
2. New 2DS has plastic on the bottom that sets it about 1mm off the PCB. DSi does not.
You have to remove the extra 2 front legs (don't simply bend them inward, this blocks cable insertion) and use something as a spacer to prop it off the PCB about 1mm. I used thin cardboard. You can compare it to the existing port before you remove it. Worked like a charm for me.