Not exactly overthinking, more like misinterpreting. Claude is not a person. LLMs don't think, want, feel, or have any kind of inner experience whatsoever. They just generate words (or tokens, to be more precise) based on a complex statistical algorithm. The algorithms tries to generate along the same patterns it saw in its training data. This means that the responses you saw are nothing but echoes of the impatience of other people, as captured in diverse forms of text (for example reddit conversations).