How effective have cognitive behavioral therapy type techniques been shown and proven to work for anxiety based on published literature?
Meta-analyses are usually pretty accurate, as they essentially bring together the data from many studies that investigate the same topic of interest together and use statistical methods to effectively get the most conclusive answer possible, throughout all of the studies assessed, just HOW effective is a certain treatment?
For your specific question, there's one in particular that strongly supports CBT for anxiety. It looks at 71 different studies (that's a ton). It also investigates it's effectiveness in a variety of different environments and versions of it's therapy. It was even published only last month.
Across 71 studies, the conclusion used the word "very effective," which is very rare to see in any study in the conclusions section, especially a meta-analysis, the weighted mean dropout rate across all different types of its application was only 15.06%, AND gains were generally maintained for 12 months (maybe more, I don't have full text) after beginning therapy.
The only caveat they mention is that the methodology and reporting quality must be improved, but I feel that's a general trait that almost all studies need improvement in.
Based on this, I'd have to say that there is very, very strong evidence that Cognitive Behavioral Therapy is an effective treatment for anxiety disorders.
In my opinion, and as an analogy, it's almost just as if not more proven than using aspirin or acetaminophen to relieve a simple headache. It just works according to the literature, which, if you have that many studies then "routine clinical analysis" really does become reality. More and more clinical psychologists are switching over to this method or attempting to make some sort of derivative of it.