- Disconnect the charger and all peripherals.
- Connect the original Surface charger directly to a wall outlet.
- Press Volume Down + Power together. Release Power when the Surface logo appears, but continue holding Volume Down.
- See whether the UEFI screen appears.
If UEFI appears, the device is at least reaching firmware, which helps narrow down the problem. If it repeatedly shuts down before reaching UEFI, a hardware power or battery fault becomes more likely.
My recommendation: Since you already have a warranty service order open, don't attempt a USB recovery or reinstall Windows yet. Neither is likely to solve a failure to maintain power, and you should avoid unnecessary risks before warranty inspection.