It is my understanding that a certain number of "JM" stamped barrels went to the NY Remlin plant in 2010, along with other components from the old North Haven plant. So that means that there are some Remlin built rifles with the JM roll marked barrel's out there.
IMO the "JM" thing is over used and confuseing to some. In about every add I read with a Marlin for sale, no matter what the DOM is, it's stated "JM" stamped. I seen one in a 1949 336 add. That's about 60 years before the roll mark was changed.
Some of the old store brand 336's didn't have a JM stamp. I have a 69 Westernfield (Marlin 336) that has a "P" roll mark. But that don't mean it's not a genuine New Haven or North Haven made Marlin.
I once thought your statement was correct about left over barrels being sent to Illion with the JM stamp, but former North Haven employees have said that the stamp was only applied to fully assembled 'proof tested' Connecticut-built rifles. Barrels sent to the new plant weren't stamped, we were told.
The stamp may not have been applied to other brands like Westernfield. It was a Marlin proof mark.
In my opinion, the vast majority of JM stamped rifles are genuine Marlins. There may be a few counterfit stamps on Remlins or a few Remlins with barrels salvaged from junk Marlins, but I think those are few and far between.
As BFPWG has stated, the serial number is the only valid test for a Marlin pedigree. If we stick with that test, the JM stamp essentially becomes a preliminary screen for authenticity and a traditional marking for the rifles we take so much pride in owning. For genuine Marlins, the JM stamp does mean the barrel was proof tested.