Sure you can: a classic XOR one-time pad has no authentication and it's the definition of confidentiality-preserving.
It's true that malleability makes it easier to attack certain protocols which can then result in a loss of the confidentiality property. Your padding oracle example is a case in point, and there's seldom a reason to avoid an authenticated encryption mode for a message that's sitting inside a TCP packet.
On the other hand, sometimes malleability is a desirable property, and authentication is harmful. The APCO Project 25 radio standard, intended for use in the mobile radio world, has encryption as an option. The priority is to maintain confidentiality and to degrade gracefully in a hostile RF environment.
Encryption is applied after error-correcting codes using OFB mode, and so effects of interference apply bitwise within the data stream on decryption. An authenticating mode would be a disaster as a single bit error would result in the loss of an entire transmission.
Even if you use an authenticated encryption mode, you still have the problem of "how does this symmetric key get shared with only the intended parties?".
Any step you take past that is a computer program that holds a secret key acting on ciphertext chosen by an attacker and thus revealing some kind of behavior to them. There have been surprising "oracles" found in a variety of constructions beyond CBC mode. Modern cryptography chooses to avoid this problem entirely.
Also, the XOR malleability issue is present in lots of block modes of operation.
Another note would be to point out that the limited attack you presented is a lot simpler to understand and execute in CTR mode (and has the benefit also of being more common in practice).
I don't think there's a significant different in ease of understanding between most of the modes. I used CBC as it's one of the most common and familiar modes, and because it's where I have seen this issue many times. I don't see CTR used in the wild very often.
I also tend to use CBC for other demos as well, so it makes things consistent.