Whether or not it was the original reason for the big-hole design, I am pretty sure that, when I was a kid (50s-60s), juke boxes did use a big spindle with big-hole 45s, whereas all the home record player (or gramophone, as they were once called) that I ever saw used a small spindle, and could usually accommodate either 45s, 78s or 33⅓ LPs. (The latter two speeds, of course, only ever came with small holes.)
The big spindle in a juke box would be sort of conical (with the point cut off), as opposed to the straight, small spindle of a home record player, and I always assumed this system was used because it allowed for a degree of imprecision in the movement of the mechanical arm that put the record in place to play. Singles, in my experience, usually came with a small hole, but with a centre that could be pushed out so that they would fit on a large spindle too. The adapters that you could buy were to get them back to their original condition. Maybe some were manufactured with big holes in the first place, but I always assumed that most of the big-hole second-hand 45s that you see are that way because they were formerly used in juke boxes and had had the original centres pushed out for that purpose.