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You know nothing of the mathematics of vector graphics, nothing of the internal structure of the Affinity applications and file format, nothing of Serif's development methodology,...

Your responses to people's suggestions for improvement or new features always focus on hurdles and barriers that you perceive.

Must you cast a dark cloud over every suggested improvement to the apps? A developer recently stated that he has plans to bring real circles (and, I presume, other conic sections) to the apps.

Correct. Quantization errors accumulate in integer format documents despite calculations being performed on values mapped into the 0.0 to 1.0 range of real numbers because the output of each blend, adjustment or filter is mapped back into the integer range. That's far more of a problem in 8 bits per channel documents than 16 bits per channel documents, though. A significant difference between Affinity's integer document processing and floating point document processing which you haven't mentioned being aware of is that operations are performed in a gamma-adjusted colour space in the former case and a linear colour space in the latter. The exposure adjustment in the 32-bit Preview panel scales the pixel values of the view composite. There is an adjustment layer named Exposure which similarly scales pixel values of the underlying document composite.

CAD software has had solutions to these problems for decades, and it's obvious from the current state of the Affinity apps that the developers are knowledgable and skilled enough to implement solutions. Of course, you cannot export a mathematical definition of an exact conic section to a vector file format that simply has no support for such a definition, so in that case you export an approximation in a supported form.

The problem is not one of precision; the four joined cubic bezier curves, which Affinity uses for approximating circles (and ellipses in general), simply cannot describe a true circle.

The macro works correctly on my machine, so I don't know where the problem is.

Of course, and I only posted the GIF to show how it is possible to approximately fit an approximate "circle" to three points in the current Affinity apps, which is better than nothing when it needs to be done using an Affinity app.

Not using an Artboard is not a bad habit. I do not know why the PDF export insists on rounding up to the nearest pixel when exporting "Whole document".

Buy a Mac.

Look at the status bar for the modifier to "create along line". Since it is ctrl on a Mac, it'll probably be Serif's infamous RMB+LMB combination on Windows.

Yes, ctrl for "along line" plus cmd for "from centre point".

Try using an Artboard. An Artboard will be exported to PDF at its real size instead of being rounded up to the nearest pixel if you choose the Artboard instead of "Whole document" in the export options for PDF. TIFF is a raster format so it will always be an integer number of pixels wide and tall.

5 page spread for a jacket with flaps for a hardcover book.

First, the Affinity apps currently have no true circles; they only approximate a circle with 4 connected cubic Bezier curves. The "circle" can have its centre snapped to a location and its path precisely snapped to any one of the three points, but, because it is not a true circle, the path will not precisely pass through the other two points.