Signal Processing answer, normal vs preview:
It's only toward the bottom of the post, yet it doesn't happen toward bottom of a much longer post. I confirmed it's also reproduced only with the offending bit, pasted below.
Reproduced in Firefox, Edge, Chrome, Windows 10, Windows 11. Reproduced on Cross Validated by pasting and submitting an edit temporarily.
<img src="https://i.sstatic.net/Gwa3Q.png" width="330">
`fft(x)[0]` is the plot summed. Note the primary signal here is obliterated by the window's excellent frequency resolution, and what remains is leakage. The image shows the `'constant'` case, with DC zeroed; its removal matters because `fft(win)` effectively behaves like `[1, -1, 1, -1]`, so every non-zero input makes a huge difference. Yet, removal of any bin adjacent to DC would have a nearly identical effect.
This explains the behavior in terms of DFT and resolves the apparent "paradox". I don't think it's possible, however, to do this for more than just one case at a time and prove dependencies on $\tau$ as we have. Sometimes the DFT has a DC, other times not, and adjacent bins also vary. It's what motivated my closed form solution.