What is the typical charge for those? I remember traveling around Italy by rail pretty cheaply.

SNCF prices vary a lot of when you book. You can see [1] that prices in the next week vary from 29 to 85 euros (I think you can get even cheaper booking further in advance)

[1] https://www.sncf-connect.com/app/en-en/home/shop/results/out...

If we could just get rid of that pesky Connecticut in the way

That is interstate commerence, amtrak as a federal agency could win any fight if they tried. But that needs leadership that cares (congress)