As others have pointed out, it is important to understand that there are three iterations of the Utah Central, each unrelated to the other. The first was the Mormon-build standard gauge line that ran between Ogden and Salt Lake City, and later was merged with the Utah Southern to reach Milford. This became part of the Union Pacific system.
In response to Union Pacific's monopoly, Mormon capitalists built a new Utah Central later on, three foot gauge, to reach the silver mines in Park City and by extension the coal mines in Coalville. This one was absorbed into the Rio Grande Western, and is the source for the 306.
The third Utah Central was organized as a switching shortline in the early 1990s and is currently operated by Patriot Rail.
Regarding the 223, Salt Lake City wanted *a* locomotive. But they had no idea what they were asking for nor any requirements for what it should have been. In 1940 when the request was submitted, the knowledge of locomotive heritages and roster changes was pretty much nonexistant, especially during a period of Hollywood-esque cowboy worship in which Utah's history was mythologized into a caricature. So, the fact that 223 was the one that wound up in Utah was purely because it happened to be the closest available locomotive to be sent over. Would have been nice if they picked a standard gauge engine though.