There are no northbound Amtrak trains that stop at Providence between 6:58 Am and 9:28 AM so people commuting to Boston from Providence during that time aren't going to take Amtrak.
On the other hand, the time I listed for the return leaves at the same time as a Northeast Regional, so scratch that. You're right. That wouldn't compete with Amtrak.
Revision: only the express train in the morning, at least until I find a better time for a return trip. Whoops.
You should see how crushed the 66 is coming out of PVD on weekday mornings, even though it's ostensibly the slowest (Amtrak) train on the line and must contend with a diesel engine and baggage car. (Despite these limitations, that train is great at beating its estimated arrival time - sometimes by 20 minutes or more.)
So it seems to me that while there certainly is going to be a market for PVD-BOS (especially in a few years when RIDOT gets their own trains going, with people transferring at PVD), there is just as much, if not more, of a market along the rest of the line. Expressing 2000 or 3000 riders from PVD versus picking up 13000+ riders along the way? And clogging up the already congested NEC by doing so? I dunno...
Here's what I come up with:
Amtrak Northeast Regional #66 from Washington DC to Boston South Station leaves at 6:58, estimated arrival 8:00 - 1 hour, 2 minutes. (Stops en route at Route 128 and Back Bay.)
MBTA Commuter Rail #808 from Wickford to Boston South Station leaves at 7:12, estimated arrival 8:16 - 1 hour, 4 minutes. (Stops en route at South Attleboro, Attleboro, Mansfield, and Back Bay.)
MBTA Commuter Rail #810 from Providence to Boston South Station leaves at 7:35, estimated arrival 8:51 - 1 hour, 16 minutes. (Stops en route at South Attleboro, Attleboro, Mansfield, Sharon, Canton Junction, Route 128, Hyde Park, and Back Bay.)
MBTA Commuter Rail #812 from Wickford to Boston South Station leaves at 8:20, estimated arrival 9:32 - 1 hour, 12 minutes. (Stops en route at South Attleboro, Attleboro, Mansfield, Sharon, Canton Junction, Route 128, Hyde Park, Ruggles and Back Bay.)
Amtrak Acela Express #2190 from Washington DC to Boston South Station leaves at 9:31, estimated arrival 10:25 - 54 minutes. (Stops en route at Route 128 and Back Bay.)
MBTA Commuter Rail #814 from Wickford to Boston South Station leaves at 9:45, estimated arrival 10:57 - 1 hour, 12 minutes. (Stops en route at South Attleboro, Attleboro, Mansfield, Sharon, Canton Junction, Route 128, Hyde Park, Ruggles and Back Bay.)
Amtrak Northeast Regional #190 from Washington DC to Boston South Station leaves at 10:14, estimated arrival 10:59 - 45 minutes. (Stops en route at Route 128 and Back Bay.)
Trains arriving at Route 128:
#66 at 7:36
#906 at 8:07 (31 minutes later)
#810 at 8:30 (21 minutes later)
#908 at 8:45 (15 minutes later)
#812 at 9:08 (23 minutes later)
#844 at 9:26 (18 minutes later)
#910 at 9:57 (31 minutes later)
#2190 at 10:03 (6 minutes later)
#814 at 10:33 (30 minutes later)
#190 at 10:43 (10 minutes later)
Trains arriving at Back Bay:
#708 at 7:54
#66 at 7:55 (1 minute later)
#604 at 8:09 (14 minutes later)
#808 at 8:11 (2 minutes later)
#906 at 8:27 (13 minutes later)
#732 at 8:35 (8 minutes later)
#606 at 8:37 (2 minutes later)
#810 at 8:46 (9 minutes later)
#908 at 8:58 (12 minutes later)
#608 at 9:08 (10 minutes later)
#734 at 9:20 (12 minutes later)
#812 at 9:27 (7 minutes later)
#844 at 9:44 (17 minutes later)
#712 at 10:03 (19 minutes later)
#610 at 10:11 (8 minutes later)
#910 at 10:12 (1 minute later)
#2190 at 10:19 (7 minutes later)
#814 at 10:52 (33 minutes later)
#190 at 10:54 (2 minutes later)
Train #844 is a weird Providence-line-that-starts-at-Attleboro-instead train. #600-series trains are Needham Line, #700-series Franklin Line, #900-series Stoughton Line. Back Bay is 3-tracks (technically 5, but Worcester Line platforms cannot be used by NEC trains) and all trains must stop there - up to two trains in one direction and one train in the other may be stopped in the station at any given time. Padding 'at one time' out to a 5 or even 10 minute window is reasonable, and that gives us a schedule hole where we could fit another train somewhere between #810-#908-#608-#734-#812-#844-#712.
Amtrak isn't using train designation #120, so hypothetical Northbound Regional Train #120 to Boston from New York Penn Station could slot nicely in there - no apparent schedule conflicts with Metro-North exist.
Such a train would leave NYP at 4:55 AM, arrive in NHV at 6:31 AM, arrive in PVD at 8:21 AM, RTE at 8:59 AM, BBY at 9:10 AM, and BOS at 9:15 AM, which looks pretty good to me - as I'm sure it does to anyone working a job in Boston with flexible enough hours to start at 9:30 or 10:00 am, or anyone commuting into Providence from Kingston (you'd be surprised) working a job that starts at 9:00.
Side Note: Amtrak should make monthly passes usable on Acela trains between Boston and Providence, or drop the fare from $34 to $26 - in line with a PVD-BOS business class Regional ticket.