I parents live within a 5-mile drive from that location. I don't think the commenters understand how windy that place is. The wood may look more sturdy but it's also a wind sail. The thin "toothpick" structure was most likely designed to allow the wind to pass through without blowing the bridge down.
And how do you propose you get all those bricks to some remote pass in Wyoming in the late 19th century? There was 21,000 (non-native) people living in an area larger than the entire United Kingdom in 1880.
It's not jist about what's readily available either. A giant brick viaduct would require thousands if workers, housing, food, and sanitation for them. On top of that, the army would have to dedicate a full company ti protecting the workers due to the risk of attack from natives. The logistics just make it impossible for such a remote area back then.