It worked a treat until about 3/4s through whereupon the auger popped out the side.
Very deep (long) holes in wood are difficult to make because of the non-uniform structure of the wood.
Wood has grain, concentric alternating structures: hard (summerwood? heartwood?*) and soft (springwood? sapwood?*). This is the reason why it is difficult to make a STRAIGHT long hole, especially when these structures are aslant and are randomly arranged in the drilled bar; this is something completely different than homogeneous metal.
So, this is almost always the case and this is the reason a very long drill in wood is pulled to the side and deviates from the axis because a very long drill bit it is not perfectly rigid, but elastic.
This is the main reason, while the second reason is the asymmetry of the traditional drill blade edges, even a small one; as you say, perhaps special drills with one cutting edge (self-centering barrel drill or cannon drill, type D) would do the trick - but I am not personally convinced of this. In addition, you will have to withdraw them every few millimeters, because such a drill has very little space for chips.
This is generally a very difficult job, you have set yourself a serious problem.
I used to make such long wooden bars with a perfectly concentric "blind"(
) long hole - but first I cut lengthwise a slightly larger strip in half, next I milled channels in the two half-strips with a ball-end cutter, then I glued these strips and only then I processed the whole thing in a round shape.
I don't know if such an option is acceptable to you.
* I don't know the English terminology regarding wood structure, I don't know if I used these words correctly.