I've seen this come up before, and I'm pretty sure (damn sure, actually) you have not a thing to worry about.
Mallable iron is a good material to build a low cost guns action from. It case hardens well, and you have a dense, unhardened core after that is over, exactly what gun builders of the era were looking for.
I'd be far more worried about the old wood if you subjected it to heavy loads, rather than the action. I've seen a few with broken stocks.
I used a 12 gauge version of the same gun for a few seasons. A nice trip back in time, but, mine had a single trigger and was a bit heavy, so, it went down the road.
Enjoy.
Best,
Ted
