Why should one not use CBC to encrypt the private keys ?
Block cipher should be favored in a use case of a wallet. And since CBC doesn't have any important downside, why not.
With 'random access' you probably mean the possibility to parallel the encryption/decryption?
No, this is not necessary since you are never en-/decrypting a lot of data.
Why should PCBC be used ?
There is a problem with PCBC.
Swapping 2 ciphertext blocks will result in the incorrect decryption of the 2 corresponding plaintext blocks.
But due to the nature of plaintext XOR ciphertext the error cancels in PCBC mode.
So if the integrity checker only looks at the last few blocks it could be fooled to accept a partially corrupted message.
That's one of the reasons why Kerberos switched from PCBC to CBC in version 5.