Have the core CBT "Thinking Errors" been found to NOT be effective in evidence-based clinical practice?
I think you should change the question a little bit to "What should be done to the Thinking Errors?" or "Just identifying and evaluating Thinking Errors might not be effective?". Yes, they may not be effective. Please be careful that all mental problems need to be treated by professional therapist and a very important part of their expertise involves the diagnosis. There are many anxiety or depressive disorders that they manifest similar "Thinking Errors" kind of thinking, but they have different underlying causes and should be treated differently. It's not to say that strategies like those described in The Feeling Good Handbook don't work, rather they work when things are simple. If you feel that things are more complicated you should surely seek some professional advice.

To illustrate this point to you by an example I would like to mention Obsessive Compulsive Disorder (or OCD). In OCD, there are some problems in cognitive system that makes the person being constantly paralyzed by lots of Intrusive Thoughts. The content of these thoughts may be similar to "Thinking Errors" or could be corrected by similar strategies, but focusing on them and trying to correct them make them stronger, more sailient, and more frequent. People with this disorder then feel that they are compulsively correcting their erroneous thoughts, but it seems they can never get rid of them. For this disorder, another treatment strategy has been offered called Exposure and Response Prevention (EX/RP). In this kind of treatment people are encouraged to face their intrusive thoughts, but don't respond to them.To try to accept their presence and at the same time not to let them to make them do anything, or not do anything. There is abundant evidence that EX/RP is effective for OCD.

And there are also other disorders with their idiosyncrasy. I'm not sure what kind of problem you're describing, but if you want to use self-help method (with all it's pro and cons) you could take a look at diagnostic features of OCD.