The problem here is not so much with CBC, but with alternatives that are easier to implement safely, without losing mathematical security.In fact, AES-CBC turned out to be notoriously difficult to implement correctly. I recall that older implementations of transport layer security don't have cryptographically secure initialization vectors, which are a must-have for CBC mode
A lot of recent attacks are padding oracle attacks, like the Bleichenbacher attack. These especially depend on old modes kept for support. POODLE is a downgrade vulnerability. LOGJAM is downgrading TLS to old, export-grade (read NSA-sabotaged) crypto suites.
For CBC mode, there is the Vaudenay attack.
These attacks depend on the server explicitly saying "invalid padding", thereby leaking 1 bit of information on each transaction. Error messages were removed, but the problem of timing remained. The server still used more time before responding if the padding was valid.
In response they were forced to come up with the peculiar workaround of generating a dummy key, and using that for decryption so it would fail in another part of the implementation. In every implementation. So they decided to no longer force that on developers by supporting it in the specs.
Cryptography is a very broad field, and a specialty on its own. History had learned through uncomfortable experience, that doing it perfectly is almost never guaranteed, even for the best in the field. For example MD5, created by Ron Rivest, co-inventor (and part-namesake) of RSA was widely used, then broken in 2013 . Its collision resistance was circumvented in 2^18 time, less than a second on a desktop computer for 128 bit hashes.