The "if it shoots modified with lead, it'll shoot full with steel" rule comes from the days of fixed choked shotguns, prior to the "screw in" chokes......I'm not so sure it's still valid.
The only way to be sure is to pattern your gun.....that will give you an idea as to how tight it's shooting at various ranges.
As for the gun, I shoot the same 870, and the factory modified tube works just fine with steel shot. I prefer larger shot (I've settled on #3, but #2 works well also). My full choke tube is rated "Full" for both lead AND steel.