I'm not going to suggest that their primary mission should be hauling PV operations. However, I do believe that there are cases where Amtrak can facilitate such moves to significant advantage for themselves. The haulage fees might not be the sole reason for Amtrak to run a train, but so long as those fees more than offset any net expenses (e.g. additional fuel consumption) there's no good reason for Amtrak not to take steps to be as cooperative with such operators in running operations on otherwise regularly-scheduled trains. Moreover, at super-peak times when Amtrak can reasonably expect to outstrip their existing equipment capabilities (Christmas, Thanksgiving, etc.) I see no reason that Amtrak shouldn't be aggressively courting such operators and working with AAPRCO and others to coordinate (and somehow give a listing of) extra capacity to deal with excess demand. Every thousand-mile car haul is worth around $2800 in revenue for Amtrak; I would presume that this rate is high enough to cover expected costs. [1]
So do I believe Amtrak's mission should cover this? As long as it can be done in a way which results in a net reduction in Amtrak's operating subsidy needs, hell yes. This isn't the mail/express boondoggle...this is something that Amtrak, to my knowledge, has more or less always done and has been more than happy to pocket the money from. Amtrak should price such moves so that they reduce Amtrak's subsidy needs and work to support and encourage such moves so as to maximize such reductions. At that point I don't think it's "supporting private car owners moving their cars around" so much as it is "cooperating with potential business partners to mutual profit". Amtrak's record on this front is a bit touch-and-go to put it mildly.
I'd also point out that if a third-party operator were able to profitably run a set of cars on the back of Train X, almost regardless of which train Train X happens to be, their ability to do so could (and should) be used to underline any mix of equipment/capacity shortages [2], demand for better classes of service [3], and so on. If Amtrak's fear is that someone might put together a business model to compete with them on those (per
their accounting money-losing) LD routes, I'd argue that they are most assuredly going about dealing with that concern in absolutely the wrong way.
For what it is worth, there are also cases where Amtrak could seek to improve "through-ticketing" revenue. There is frankly no good reason that Saratoga and North Creek shouldn't be in the timetable (or any other co-located tourist train that goes somewhere) and/or available for through-ticketing (e.g. a ticket which is NYP-SAR on Amtrak and then SAR-North Creek on the Saratoga and North Creek). Long-term I'll grant that All Aboard Florida, Texas Central, and whatever form CAHSR ultimately takes may be a slightly different kettle of fish...but working to co-list services in exchange for nominal referral fees would seem to be worth it for both parties. My understanding is that Ed Ellis would absolutely LOVE this and that the hangup is on Amtrak's end.
This is to say nothing of potential partnerships between Amtrak and Greyhound...though in this particular case, things are getting better (Amtrak seems to slowly be adding more Greyhound buses to their Thruway network, for example). Even if it were simply a matter of cross-listing non-contracted connecting services going places Amtrak doesn't (e.g. Jefferson Lines from Osceola to Des Moines comes to mind) it would certainly help with major connectivity gaps in the system. Look over at Swadian's posts and see just how thin Greyhound's network is getting...and the two companies often only vaguely cover the same cities (e.g. Greyhound's Chicago-Denver service stops in Omaha as well, but their routing is basically the ex-Rock Island/Union Pacific routing rather than Amtrak's ex-CBQ routing) and in some cases, such as Atlanta-Florida, there is functionally no competition between the two (yes, you can book ATL-JAX on Amtrak, but you end up going in a rather "special" way to get from A to B which is going to run off most non-railfan travelers).
[1]
http://www.amtrak.com/ccurl/892/360/Private-Car-Tariff-Rates-Addendum-3.pdf
[2] Which are manifest throughout the system.
[3] Something which also stands out in many places. The fact that there is generally no "upgrade class" available beyond a highly mediocre "Business Class" even on some very-long-haul routes (and not even that on at least one route) seems to me to be facepalm-worthy. That the BC offering in question is usually a surcharge of about 25% versus coach pricing is actually rather stunning considering what "First" or "Business" on an airline will often fetch.