To close this up: 

I went to an Apple Store had the logic board replaced. Afterwards I reinstalled from a backup and everything seemed fine... 

But yesterday I noticed that I forgot to reenable disk encryption (using File Vault). After enabling it and rebooting I got the exact same problem again. This time I also experienced the problem described here: [Password incorrect after enabling FileVault][2]

But unlike last time the problem disappeared (so far) after disabling encryption again. 

**If you wind up here looking for answers:** Try disabling File Vault and check this for more details: [MacBook startup problems - mouse and keyboard not working at login / stuck on apple logo][1]


  [1]: https://apple.stackexchange.com/questions/262964/macbook-startup-problems-mouse-and-keyboard-not-working-at-login-stuck-on-ap
  [2]: https://apple.stackexchange.com/questions/268595/password-incorrect-after-enabling-filevault/269035#269035