If Amtrak runs it (I think this is the only real way that it would ever reach Concord again), it ought to run all the way to Montreal.
Who has higher fees, MBTA or Amtrak? I am assuming Amtrak charges more for crew, cars and locomotives. CTDOT chose another carrier for their Harford Line service while owning or leasing the equipment themselves. CTDOT contracts with one company to run Hartford Line while contracting with Amtrak to run Shore Line East and Metro-North to run New Haven Line. Different decisions based on cost, convenience, etc.
As for frequency, Amtrak would run whatever frequency NHDOT wants to pay for, assuming they have the crew and equipment which they probably don't. No reason some trains can't run to Concord and others terminate at Manchester. Amtrak also doesn't have maintenance facilities on the north side and would have to shuttle everything to the south side or rotate out on the NEC or Lake Shore Ltd.
NH's 2014 document says NH would have to pay for 1 eight-car? trainset to be added to the pool. A shuttle operation like I suggested would probably need 2 shorter trainsets + spares. The MBTA could refuse to allow another provider to access Lowell station or tracks in MA, but imagine how bad that would look. They could refuse service, saying Lowell platform doesn't have room for multiple trains to dwell, or isn't long enough for trains to dwell head-to-head or the train blocks don't allow it or some other bologna.