It's Ithaca themselves that is feeding that "line of BS."
OP, that's correct. Not all Ithaca M37 barrels are interchangeable, and not all barrels that are interchangeable end up with the correct head spacing. It's not how loosely the barrel turns in the receiver that creates the issue, it's how far the barrel extension extends into the receiver. After all, the barrel doesn't screw into the receiver on an M37. It slides in then turns to engage the locking threads. The bolt locks in top of the receiver in a M37, not into the barrel extension like many newer designs. Headspacing needs to be checked, especially if you are mixing and matching receivers and barrels from different years.
In fact, some newer M37 barrels won't fit some old M37 receivers at all without having the receiver milled and having an insert installed to accomodate the newer barrel's thread pattern.
I recommend you deal with the gunsmiths at Ithaca. Their service is stellar. The best I've experienced with any company or individual. I've used them several times and they will work on all years of M37, whether they made it at their factory in Upper Sandusky, or it was made at Kings Ferry or Ithaca.
https://ithacagun.com/factory-service/BTW, if you do have a barrel that does fit the receiver properly but turns in tightly, this is usually due to the barrel yoke split that is mounted on the magazine tube being too tight for the barrel. This is because the tighter the barrel yoke split is, the more upward pressure is exerted on the barrrel. If you try to attach another barrel with the yoke split too tight, then you are essentially attempting to put the barrel in at a slight angle. You can loosen the yoke split screw a bit and the barrel will turn in freely. Then you can adjust the yoke split after the barrel is in. There are slight variations in the outisde diameter of barrels - like when going from a slug barrel to a vent rib barrel, or from a rotoforged barrel to a newer designed where all the went rib posts are machined as one part with the barrel.
The yoke split need to be loose enough to turn the barrel in the receiver easily, but tight enough so that when you back the magazine cap out to secure the barrel, that yoke split doesn't twist on the magazine tube or allow the barrel to twist while attached to the receiver. The way to test it is to install the barrel, make sure it turns in the receiver easily, if not loosen the yoke split a little, then tighten the magazine cap to the barrel stud, then tighten the yoke split screw, but just enough to make sure the yoke split can't be grasped and twisted on the magazine tube with the barrel installed, ore the barrel twists while in the receiver and on the magazine tube. You may end up with a siutation where the barrel is a little tighter to take off than it was put put on.
The M37 is a great gun. But it has some excentricities. It's not the same as the mass produced shotguns made today. It needs to be tuned a little each time you change a barrel out.