That is a really nice M37. I have a 16ga from 1959 that is looks just like it.
Unless you really like a vent ribbed gun the plain barrel should not be much of an issue. They just seem to point so well.
The tang sight sounds like the previous owner might have shot a slug or to out of it???
Something you might not be aware of. The guns of this era barrels do not inter change. So I might consider opening up the choke, I would do this before I cut down the barrel a couple of inches for tubes.But that is just me.
The thing is the gun is yours do what you like.