Railroad historian here. The apocryphal Roman chariot story actually has a lot of truth to it. To summarize: Roman chariots had a particular, standardized wheel track. This was designed on the basis of the width of the rear of the two horses pulling the chariot. Since these would tend to rut roads (sometimes accounted for in road design in what is arguably the first "rail" roads).... other wagons found they needed to near-match, or the side forces of the ruts would tend to break wheels or axles. So this became a quasi-standard, in turn defining wagon widths.
Which then flowed into wagon design on early tramways, which informed early Railways particularly George Stephenson's successful efforts. There had to be a "standard" track gage, as per frickin' usual, the "popular" beat the "technically superior" (6' was a serious contender and even 7'/2.1m was tried). But here we are at 56.5" or 1435mm.
Loading gage (width/height) was the issue
Of course, track gage is barely the point; it's Loading Gauge that matters, and one does not dictate the other; witness the shockingly small London Tube loading gauges compared to the American West "Plate H Plus" sizes, that handled double-stack containers with very little tunnel modification. But still, track gauge defines a practical limit to Loading Gauge. For any speed, tracks must be superelevated (banked) in curves, and a train must also be able to stop and then handle starting drawbar force in those curves without falling inward, which does in fact happen when an engineer is not careful.
The selection of Thiokol in Utah was another case of popularity beating common sense. Anyone along a coast or navigable river could have shipped a complete 590-ton booster by barge, which would mean field assembly would not be a design requirement. Sites along inland flatlands could have at least sidestepped the diameter issue, by shipping via tunnel-free rail either to a marine terminal or simply rail following along the Missouri, Mississippi or Gulf Coast without a tunnel. Utah, however, is a mountainous bastion. So segmentation and rail tunnels were in play.
But loading gage controlled length due to curves
The segments are quite short for rail cars, which leads into a characteristic of curved tunnels. Curved tunnels must be wider to accommodate both inward swing of long-between-bogeys cars, and outward swing of the tails of long-overhang cars. The largest car generally in circulation is the 89' auto racks or hi-cube boxcars. Since shorter cars have less chord, this allows extra diameter at the cost of less length per segment, which means more segment joints. SRB segments are - well, I couldn't find a positive reference, but a bit shorter than an old 40-foot boxcar (those may be 50's).
Further, in the mechanical age, UPRR was overhauled with generous tunnel clearances, as were most in the west. They are larger than tunnels in the east. And the UPRR routing from Ogden to Nebraska along I-84 and I-80 has relatively few tunnels, and none needed beyond that point.
But yes, it's fair to say the Thiokol booster design was controlled by loading gage (particularly due to curves necessitating short loads) which necessitated the number of segment joints... and that was a bit influenced by - back to the start - a couple of horse's asses.
Namely the guy who picked 56-1/2" rail gauge over wider options, and the guy who really wanted the boosters made in Utah :)