Westbound connections at Chicago tend to be much less fraught than eastbound ones. One reason is the eastern trains simply do not have as much distance and time in which to get messed up. Personally, I almost always build in an overnight layover in Chicago heading east, but never do heading west.
With that said, since January 1st the Cardinal has missed the the Zephyr on just 4 out of 99 arrivals. The Capitol has missed the Zephyr just once out of 225 arrivals.
Your odds of making a same day connection from either train are very good. In terms of advice, all I can say is that I book same day westbound connections at Chicago myself.
In the unlikely circumstances there is a miss, you will be accommodated on the next day's Zephyr. You will get the closest available accommodation to the one you booked. That could be a bedroom, but could be a roomette, family room, or accessible room. If no sleeping accommodations at all are available, you would be downgraded to coach. They will refund the fare difference for any downgrade. They will not put a reservation holder on that train out of their room to accommodate misconnects from the previous day. The consists are set and they do not add cars for misconnects. They don't have any extras.
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