OK, I’ll stick my neck out…
The evidence suggests that a little eccentricity is (almost?) universal. Since I’m sure the reputable manufacturers are capable of grinding to better concentricity than the saws typically exhibit, one has to conclude that the eccentricity is deliberate. Why might this be?
Most of us will be familiar with a slitting saw wandering off the straight and narrow, even when we’ve done our best to get its alignment spot on: for some reason, it gets deflected sideways. It’s clearly vital that a saw starts its cut correctly, and without deviation or sideways deflection. My suggestion is that the eccentricity serves to relieve periodically the cut on the saw, allowing any sideways deflection to relax, and the saw to straighten its kerf, as the saw starts its cut, perhaps analogous to pecking with a fragile drill bit, if there’s no centring dimple. Once the saw has got properly going, I can’t think that it serves any useful purpose, except, perhaps, to allow coolant to get to the action.
[Edit. Wow, that Bold font really is bold, isn’t it (at least on my system)? Didn’t mean to shout…]
If this hypothesis is correct, the saw should be allowed to cut intermittently, at least until it’s established in a guiding kerf, rather than cranking it into cut, at the start, with a too-high feed rate.