NC is interested in having the Carolinian run beyond its state borders because I suspect the state wants to have the connectivity from NC to Virginia and Washington DC, at least, and, to a lesser extent, the rest of the NEC. For the same reasons, they **might** be interested in having some sort of Raleigh to Atlanta Service, or maybe even extend the Carolinian to Atlanta so that Rocky Mount, Wilson and Selma-Smithfield can get connectivity to Atlanta, but this increases the chance of delays on the extended route which reduces the utility of the train to passengers are just traveling between Charlotte and Raleigh.
I would think that the State of North Carolina has no interest in funding part of a New York - Mobile/ Montgomery long distance train. Such as train would have to be an overnight train in any event (the current Carolinian takes almost 14 hours to get from New York to Charlotte hours just to get to Charlotte. A New York - Atlanta/Montogomery/Mobile train would definitely need a different schedule as well as sleeping cars and food service. It would probably also nerve most of the North Carolina stations at a inconvenient hour. That's why I think that any day service to Atlanta via Charlotte is probably better off terminating on the north end and Washington DC or Richmond, or Raleigh.
I can guarantee NC people want a NY train. I don't know if NC was early in the 750-mile game, but it purchased rolling stock, unlike VA, as far as I understand it.
Outside of train business, Raleigh has vied to be and is part of the NEC. It invested in the Research Triangle orders of magnitude above anything VA did. I've heard research people (who keep a keen eye on grants etc.) make the comparison, and paint the scale of VA's efforts as more like the booster-ish economic development you have in many places. University builds a business park, etc. City opens an incubator in a storefront. Typical stuff. By contrast the Research Triangle around RGH was a world class investment and it worked. Then came NoVa and the snowballing of the multiplier effect of Beltway Banditry, as it's called (federal contracting), and perhaps the "DMV" (DC/MD/VA) has taken a march on the Triangle in "tech." The best hospitals in the region though are still Duke, Johns Hopkins and UNC.
In any case, there are only two trains a day for RGH on the NEC, and one is from Florida. And they're scheduled backwards if you're in the NEC and have business in the Triangle. Two nights in a hotel, in other words. On the other end of the state is Charlotte, an actual big city, like Atlanta. But connecting to Georgia is not as easy as being in the expanded NEC.
Supposedly slots on the Long Bridge ALX-WAS are full up until the new bridge opens about 2031. (But as has been said, maybe not at night?) One NECR does terminate at Richmond. Hooking that up to RGH is a non-starter because VA wants trains at RVM, which doesn't connect currently to RGH. And there's a conjunction what's your function problem, like people have said, until the S-Line opens from Petersburg to Raleigh (still in preliminary engineering).
Generally, the economy moves quickly now, and this slowpoke approach to building committed plans is counter-productive.