A big part of the quality of a drill bit is determined by the grind of the tip. Not only should it have the correct angle for the material you're cutting but especially in symmetry. Cheap drills are not necessarily cheaper because of what they're made of, but by how they are made.

Most drill bits you'll find outside of specialists are ground for steel, so that's ok; are you can test for symmetry by looking at the end, with a lens if necessary. If it's not symmetrical it will cut more on one face than the other, and if there are any differences in pressure or missalignments the bit will go off line. For small drills you can also put it in a chuck held in your hand - or even take it between your thumb and forefinger - and drill into some softish plastic; you watch the chips to see if they are equal, and feel how the bit is cutting.

If you have a choice, for your pilot hole pick the one with the best point.

If you know how to grind bits (not that difficult, takes a bit of practice) you can cut one down to almost no flutes and put a point on it and it use it to start the first part of your hole, it's a bit better than holding it by the corners of the flutes as in the picture; stiffer, and you're less likely to add to the bit's woes with over-zealous tightening of the chuck jaws.

A trick for centre-punches, especially if you have to point one by hand: make a triangular point. Make all the faces look the same and you know the point is on the axis.
A drill bit with two lobes won't wobble in a three-pointed punch-mark, and I find it easier to see if the point is on the scratch-mark.

The quality of the hole is determined in great part by fixturing. I am impressed by your willingness to attempt and subsequent success at this tricky job holding the work and the drill-motor as you did. You could have made a simple drill-guide from a thickish bit of extruded L-section aluminium and some way to clamp it to the dropout.

Thickish enough to guide the bit in alignment not just location. Extruded because the inner corner will(should/must) be sharp enough to allow it to be clamped correctly to the dropout face. Angle-iron would also do, you just might need to dress the corner to fit.

You drill a hole in the other leg of the L one-half the dropout width away from the mating face (an easy way is clamp it to the other dropout and using that adjuster hole as a guide drill from the axle end).