For the new generation, check the ziff connector to the back light of the lower lcd. These can get loose. Also check which displays flashes when the pop sound is heard. This can be an indicator on which cable is the problem. The older generation is usually the 3d cable that had the slight tear near the hinge

If the battery is fine, check the F2 fuse

The motherboard may be in Kiosk or display mode. This only happens when the motherboard automatically turns on when it is plugged in without pressing the on button.

The other cause of blue light is a non-functional top lcd, non functional bottom lcd, or camera ribbon. Try disconnecting the lcd ribbon and see if the system boots up. If the motheboard can stay power on without any ribbon connected, then there is an issue motherboard (hardware) or with software preventing to boot properly.

If replacing the charge did not help the charging issue, then I would use a multimeter and check I have conductivity from the the port to the EMI square located to the left of the port. I sometimes find that the fuse (F1) is blown or that the EMI square is cracked, loose, or broken

If you are still having problems, try in removing the board from system. Attach the board to the battery via the bottom case and try to turn it back on. If the blue flashes and turns off, then you know the board is working and the issue may be one of the ribbon cables or the LCD. If the blue light stays on, then issue with the board

Check the F2 fuse to see if its blown

Removing the wifi card resulted in a positive response. Yes, the logic board will turn on. Fuses will only impact on charging the battery. I assume the L2/L4 inductors are fine because the system actually turns on. The logic board powers and see that the wifi is installed. The next step is to power on the top and bottom lcds. If there is a disruption of power to any of the lcds, then it would cause the system to power on and immediately power off. The issue has to do with the lcd ribbon cable (or connection). The ribbon cable can only in as much as it allows. When inserting, push in as far as it goes until the flared ends on each side of the flex cable touches the edges of the connector on the logic board. Make sure that you then lock the connector. Over 80% of these issues are related to the top lcd. The top lcd has to be twisted around the hinge of the unit. This, over time, (assuming that the hinge is not broken and top half is loose) may cause a slight tear. Buy a replacement lcd and try connecting it...

Check the F2 fuse next to the battery contacts (for the 3ds it located underneath, other side of the logic/mother board). If the fuse is blown, then then system will charge for 10 sec and then stop.

The 3d flex cable has to fold and face its connector. Check if the gold contacts are facing up when installing the ribbon cable. Installing them upside down (gold contacts facing down) will your system to not power on at all.