Hello - two years later, hope you still have the computer lying around. My laptop has the same symptoms and we have been able to get it to turn on by manually resetting the PRAM (a separate battery that has to be manually reset by opening the computer) each time.
The theory is that this is due to BIOS issues, and that BIOS updates could theoretically help. I ran out of patience while trying to update the BIOS, so I am not sure if i ever fully tested this theory.
If you just want data recovery, you can buy tools on amazon that allow you to plug your SSD into a USB. I’m ordering one now to try. Your friendly local computer repair shop has the same tool.