This is really a comment to Thucydides' answer, but I couldn't fit it in, or make it clear without paragraphs.
Re "Most of what you describe would mean "The Great American Desert" would not exist, so there would be no traditional tall grass prairies." That's not really the case, since that term is used to describe the area EAST of the Rockies, which would still exist in the OP's scenario. See e.g. https://en.wikipedia.org/wiki/Great_American_Desert
Since the Rockies would still provide their rain shadow effect, I THINK* that the Great Plains would be much like today, though perhaps a bit wetter. However, it's not really lack of water that made them grasslands, it was grazing by herds of tens of millions of buffalo (and their Pleistocene predecessors). Grasses and many forbs can be grazed to the ground, and will grow again from the roots (just as your lawn needs regular mowing). Few trees can do this, and those that can will remain as shrubs if subjected to regular grazing.
The area west of the Rockies would not be like the area east of the Mississippi, either. Most of the West has a Mediterranian climate: rain/snow fall in winter, while summers usually see only the occasional thunderstorm. This is driven by general atmospheric circulation & the Pacifc Ocean, so wouldn't change much. You'd still get the same total precipitation, but it'd be spread between the coast & Rockies. The general climate might be rather like southern France or Italy.
You'd also have quite a bit of rain shadow effect from the Great Basin ranges (many of which aren't all that much lower than the Sierra), and from the elevation of the continent itself. Remember that the valley floors in much of the Great Basin are mostly at 4000-6000 ft (1200-1800 m).
As for the large endorehic lakes like Lahontan & Bonneville, it's surprising that their existence wasn't because of higher rainfall, but cooler temperatures leading to lower evaporation.
WRT human settlement, note that most people didn't settle in the intermountain west/Great Plains until well after the Civil War. They passed through on their way to the gold fields of California (and later isolated mining camps elsewhere, e.g. Virginia City), or fertile farm lands in western Oregon/Washington.
*Absent a good climate model, this has to be just educated guesses :-)