The buttstock doesn't even look like it's Ithaca. Like you suggest, it looks more like a Stevens product. I can't read the buttplate. The transverse screw grooves are definitely afterthought and the full/cyl are over stamped indicating a change in status by persons unknown at a time unspecified...

The thing about guns like this is, the originator sometimes doesn't tell a good story. What he meant to say and what's understood are sometimes vastly different. This is just like the one I carried, I carried an Ithaca like this, turns into "I carried this one." And it could have been made up forever ago, before anyone thought of faking. No malice meant, just put together. Just because the current owner thinks the originator owned it before doesn't make it so. Just an accumulation of parts...

But...I could be wrong...