No, they aren't a summary. They are the actual decoding of the sequence of tokens emitted during the the “thinking” stage of response generation.
Just as with, say, a human onner monolog in words vs actual speech, they are a product of the same output process as the non-thinking tokens. They aren’t a translation of the internal process that precedes the output mapped into language, either as a full result or a summary.
Having access to the reasoning text and output would help with performance measurement.
For daily use I actually like the reasoning summary to be brief/quick to scan.
That said, I understand the author’s desire for the real thing. It just feels better to have that access, especially when Anthropic will give it to you, but encrypted.