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.
The number of rail bridge collapses covered by Well There's Your Problem where the episode opens with a retelling of how the railroad company owner designed the bridge himself because the engineers couldn't grasp the genius of his vision is too damn high.