Thats not possible. Heres why.. Chrome OS, wont recognize the way chrome stores files on a mac or a pc. While it would be possible to move your data folders from pc to pc or mac t o mac. You cant move browser data from Mac to chrome or PC to chrome. They are entirely different operating systems altogether. And use completely different file formats.
One reason that could happen, is if you chose reset sync at some point from within chrome.. T hat would wipe out your synced data. ( but wouldnt wipe out google drive since thats not part of the chrome sync service itself. )
Another reason could be if on the new chromebook you dont have the right options turned on in advanced sync settings.
again.. Most likely this is one of two issues. On the broken chromebook you didnt have all your settings set to sync.. ( if you only had one chromebook and no other pcs or phones, that would make sense as what would you sync with?)
Or B, at some point when setting up your new chromebook you went into the chrome options and selected reset sync. Which would delete your info from the server, but not from the broken chromebook itself.
The only way to really know for certain what has happened is to use an HDMI cord and connect an external monitor to your broken chromebook so youll have a way to actually see whats on it..