The routes that have the greatest chance of survival are those routes which have a rider demand for the route.
The problem, which can easily be found if you view any of the available financial reports in the Press & Media section of Amtrak's own website, is that its costs far surpass its revenues, no matter how you spin it or look at it.
Amtrak's loss is just slightly more than its labor cost, which is its single largest expenditure. Amtrak makes outrageous contributions to Railroad Retirement, yet why are many of its employees covered by RRB and not social security and a regular 401(k) plan (particularly those employees who are not in operating, or traditional railroad jobs)?
Amtrak operates passenger trains with dozens of empty seats - why not run one or two fewer coaches on its trains (which means less equipment maintenance)?
Amtrak owns 80 stations - no airline owns their own airport.
Yes - there are certain fixed costs that are system wide that if part of Amtrak were privatized, those fixed costs would have to be duplicative (i.e. reservations centers, maintenance facilities) - but then again those costs can be minimized. For example, the state of Oregon already owns and operates a reservation call center for both its state parks, as well as Washington State Parks. So, if Amtrak Cascades were to go entirely to those two states, there is already a facility that could be used, with minimal cost.
Can a passenger train make a profit? At least an above-the-rail profit, and maybe a small contribution to infrastructure, I truly believe so. Can Amtrak make a profit, even above-the-rails? I don't think so. Yes, there have been varying reports that the Empire Builder, the Acela Express and Auto Train can, and I know some of the 403(b) supported trains do after the state subsidy is included (because for accounting purposes Amtrak considers the state subsidy as income). But I don't think Amtrak is capable of such, until it reduces its internal costs.