A sudden power outage could corrupt the Mac hard drive and make the Mac unbootable. To save the unbacked-up files from the corrupted Mac hard drive, you can try professional third-party data recovery software.
Given that the Mac internal hard drive is the startup disk by default, it locates the operating system and stores user data. When Mac's internal drive is damaged/corrupted and it can not load the OS, then MacBook won't turn on and you can not access all the contents kept on it.
As you said, you have backup up the Mac hard drive before it turns to be unbootable, then you can easily restore those files with your backup such as Time Machine, iCloud, etc. For the newly added files that haven't been backed up yet, you can recover them with data recovery software such as iBoysoft Data Recovery for Mac, a professional, risk-free, reliable, and easy-to-use program.
Given that you can remove the corrupted Mac hard drive and connect it as an external disk to another Mac, you can directly download and install iBoysoft Data Recovery for Mac on the healthy Mac, select the corrupted Mac hard drive, and click Search for Lost Data to start freely scanning the drive, then freely preview the recoverable files, and check the box of wanted items and click Recover to save them. Only a few steps away from recovering your data!