The "slop" has always been there and it's not a deterrent to accuracy. The reason the 870 seems "tight" is it uses two slide bars and the mossbergs only has one. There are many other reasons to like or dislike a particular shotgun but the slide action shouldn't be the only one. I would be concerned with the placement of the safety, or the size of the trigger guard, or the sights that are on the gun, or the stock fit long before I would concern myself with the feel of the slide when it's racked. Do try the triggers out as they very not just between makers but between individual guns of the same model. But like Tom said that's an easy fix.
This is kind of like when guys say Ruger Over and Unders are loose. They aren't loose...they have a spring assist open so they fall open easily rather than having to break them across your knee. Once locked up the action is very tight and in some ways stronger than many other shotguns.
I grew up in Illinois where all you can hunt with is a shotgun and I've seen very accurate representations of both guns. (The most accurate I ever saw was a Tar Hunt bolt action shotgun.) You really can't go wrong with either gun. Try them both out and seen which safety you like better.
I don't really have a dog in this hunt because I shoot a SBE but don't let the slide action lead you to believe the build quality it less. It's just different.
The reason the 870 seems "tight" is it uses two slide bars and the mossbergs only has one.
If by slide bars you are referring to the connections between the forearm slide and the bolt inside of the receiver, my Mossy 500A has 2 slide bars. I haven't had my 870 apart nearly as much as my Mossberg (had the Mossberg a lot longer) so I don't remember how that one functions.