It boils down to “does moving to the central position gain more voters in the end or not?”

I certainly don’t think it has to boil down to that. Your question is valid but there are plenty of other questions you could derive. One I like might be: “Does ignoring the voice of your voter base increase apathy or not?” or “How does apathy caused by unquestioning support for genocide interact with other alienations caused by democrats courting conservatives?” or really poignantly “Might not the ‘left’ adopting conservative framing and policies simply reaffirm conservative leaning swing voters in their decision to vote Republican?”