>there are a few entire states Amtrak doesn't serve or run through
Actually, there are two states with no scheduled Amtrak service at all: Wyoming and South Dakota. That doesn't mean there's *good* service elsewhere (in several places it's 3 trains a week each way), but there's at least one Amtrak stop in every other state in the lower 48 (obviously, Alaska and Hawaii and the "Commonwealth" areas are out of the picture). Yes, Sandpoint ID is a long bus ride from Boise (assuming there's a bus that connects - note that a key feature of California Amtrak is the coordinated state-supported connecting bus service that makes a lot of towns that the train can't serve accessible to it), but it's an Amtrak stop. Likewise, Memphis TN (an Amtrak stop) is a long way from Nashville or Chattanooga or Knoxville, but it's in the state.
Source: Amtrak Live Map
Just for fun, I looked for how many state capitols are served by Amtrak, and the closest stop to those that aren't. Here's a guess based on the Amtrak Live Map (OpenStreetMaps base) and 50states.com. I didn't have time to dig out the GIS and download a bunch of data from various sources.
Served directly (27):
California (Sacramento), Oregon (Salem), Minnesota (St Paul), Michigan (Lansing), Maine (Portland), Vermont (Montpelier), Massachusetts (Boston), Rhode Island (Providence), Connecticut (Hartford), New York (Albany), Indiana (Indianapolis), Illinois (Springfield), Nebraska (Lincoln), Colorado (Denver), Utah (Salt Lake City), Kansas (Topeka), Missouri (Jefferson City), West Virginia (Charleston), Pennsylvania (Harrisburg), New Jersey (Trenton), Virginia (Richmond), North Carolina (Raleigh), South Carolina (Columbia), Georgia (Atlanta), Mississippi (Jackson), Oklahoma (Oklahoma City), Texas (Austin)
Nearby station (suburban or within about 50 miles) (9)
Washington (Olympia, via Lacey), Nevada (Carson City, via Reno), Arizona (Phoenix, via Maricopa), New Mexico (Santa Fe, via Lamy), Wisconsin (Madison, via Columbus), Maryland (Annapolis, via BWI), New Hampshire (Concord, via Durham), Iowa (Des Moines, via Osceola), Louisiana (Baton Rouge, via Hammond).
The others (10) appear to be more than about 50 miles from an Amtrak stop, which is not necessarily in the same state (Columbus, OH, for instance, looks like a near-tossup for Cincinnati OH or Portsmouth KY as the closest stop). As noted above, 2 in the lower 48 states have no scheduled Amtrak service at all.
Are state capitals "rural?" Some probably are by big city standards. But not all. And at least 12 have no convenient Amtrak service, if there's any in the state at all.