No, all TLS CBC cipher suites have the same problems: most old implementations and a minority of currently deployed implementations (some unpatchable, some where the vendor could not be contacted by researchers, etc.) have a variant of lucky13 or poodle (both have many named variants).
The Mozilla list is likely from before the discovery of the new variants of poodle which finally caused SSL Labs tester to mark all CBC cipher suites "weak".
It is possible to implement CBC cipher suites correctly, but it is too hard. If the effort is made to implement them correctly, it would be better to spend the effort at supporting GCM or CCM or ChaCha20-Poly1305.
If the other side can support AEAD, you should use AEAD. If the other side can't support AEAD, you have to be very suspicious and test its CBC thoroughly. TLS is supposed to be secure without first testing the other side, so if you can drop CBC support you should do that. This means you drop support for TLS versions below 1.2.
Of course, for business reasons, many will choose to support CBC cipher suites for a long time. When can a business drop support? When no such clients exist in practice. When will users stop using clients (or proxies) that can only do CBC? When such clients can't connect to important servers (e.g. google, facebook). This will take a long time.
Chrome might show UI for "insecure" for sites that require CBC cipher suites next year, but dropping them completely - I guess not. Google's servers can't show UI for insecure - they either allow connections or not, so I think they will allow connections with CBC cipher suites for a long time (though they have published they want people to upgrade to AEAD and only promise to support ECDHE with AEAD).