No wonder the replacement packs are expensive. That's a complex battery controller PCB. It's custom, so you aren't going to find an off-the-shelf replacement. It uses an off-the-shelf TI BQ24296M charge controller
https://www.ti.com/product/BQ24296M, a low-end STM32 ARM micro-controller
https://www.st.com/en/microcontrollers-microprocessors/stm32f030f4.html, and some other circuitry, at least some of which appears to be a small buck or boost converter. Looks like the STM32 controls the LED.
I'd first take a close look for things like a bad/loose solder joint, bad wiring harness connection or corrosion somewhere that might be causing enough of a short to cause a failure. Then things you might start with to debug this PCB are:
1. Check VDD on the STM32 (pins 5 for analog power in, 16 for digitial power in). I guess you should see 2.8V, 3.0V or 3.3V. Ground is on pin 15. Failure to see good power would probably implicate one of the parts above the micro.
2. Poke around the BT24296 chip. While charging pin 4 (STAT) should go low (the designers helpfully pulled it high even though they don't appear to use it using the 3rd resistor above the cap on the lower right of the board). Pin 3 (PG) should go low if the chip likes the battery voltage (2nd resistor above the cap).
3. Use your scope to probe the SW out signals (BQ24296 pins 19, 20) going to the inductor at the bottom left to see if it's trying to provide power to the system - or you could measure across the two caps above the inductor (according to the spec sheet this should be 3.5-4.35V).
4. If the charger seems alive and the micro is getting power you could probe the I2C bus between the charger and micro (the 4th and 5th 10k resistors above the cap on the lower right). This might show code is running on the micro.
If it's the BQ chip then that's an easy replacement. If it's the micro then you're probably SOL. It could also be one of the parts above the micro but I didn't really try to figure that circuitry out. Other parts you could also probably replace although you'd have to do that dance where you figure out what part some small chip is based on the cryptic markings on top...
Would be curious what voltages the battery pack puts out and what communications it supports to the level. Sure is a complex pack to just house a 3.7V battery (your two cells are wired in parallel).