What is the name of an attack where the compromise of one key compromises the whole system?
The whole system depends on the security of one private key (..that gets..) compromised.
  Give a suitable name for such an attack.


I would call this class of attacks: Master key compromise.



Examples of such attacks (as asked in comment) include


The master key is revealed to an attacker by physically stealing a backup copy, bribery or duress (obligatory XKCD) of a holder of the master key. Mitigations use safes, and/or k-out-of-m secret sharing.
The adversary extracts the master key from a device holding it (the most obvious example of that is when the master key is in every device, which is not unseen). Mitigations include using security-certified device at least for master keys (HSMs, SAMs, Smart Cards). Method of extraction include


directly reading the memory/medium that stores the key,
examining memory where the device using the key copies it, perhaps temporarily (e.g. with a debugger, a JTAG port, exploitation of a buffer overflow vulnerability somewhere in the code..)
timing attack, (differential) power analysis and similar side-channel attacks (including fault attacks)

Cryptanalytic attack or/and brute force attack, where a secret key (including private) is inferred from public material, like ciphertext, plaintext made available to an adversary, public key. Mitigation include using larger security parameters for master keys.