I'd desolder the burnt ic and check if there is a short across the 16v caps. if there is no short power it and check if any of the 16v caps has any voltage on them. if theres no voltage on any of them also remove ic8. if you have a lab psu set it to 5v and current to 0.1 amp and inject it to the 16v caps and if you're lucky it powers up. if not slowly turn the voltage up until something happens. if it's working get a buck converter and solder the input to the 35v cap set it to to the same voltage as the psu and solder the output to one of the 16v caps.
id say 5v should do, as those caps go directly to ics. if both, ic7 and ic8 work independently from each other probably need to inject 5v to at least two of those 16v caps.
Most usb- c pd adapters wont put ouz any voltage on their own. the end device must tell the adapter what voltage they want via the cc pins. on the raspberry pi this is done via resistors between cc1 snd cc2 to get 5v. those pins are most likely unpopulated on your bycicle lights, so they get no power.