I've owned and shucked more than a few pump guns and never understood the focus on action bars as the source of gritty notchy movement. ......
I think it's the fact that the bar(s) are readily accessible, you see the parts moving back and forth,,so it makes people feel like they can improve things by working them over.
No doubt some improvement can be gained,,any small amt of reduced friction or resistance helps.
But the majority of any roughness or hard operational effort felt is within the trigger guard mechanism, hammer & carrier and it's springs unless something is out of wack to begin with.
Take that 870 and TD it.
Remove the bolt & the bolt slide from the action slide and slip the forend assembly back onto the gun. Try the pump action again and see how it feels now.
Works pretty easy w/o the bolt & bolt slide linking up with the hammer and working the carrier, disconnector, springs, ect in the trigger group.
There is much more to the resistance and feel, or lack of feel of a smooth action than those action bars.
The 31 is just smoother to operate because of the design of the carrier (2 narrow rails for the bolt to slide on) and the way the bolt tips it up for feeding.
It's similar to the Rem 17 and the Ithaca 37 style carriers,,both known for slick actions.
The Mossberg 500 copied it to an extent also.
Twin action bars,,,, nice I guess but if the gun is designed and built right they aren't necessary. Plenty of well known pump guns have been around for a century+/- with out 2 of them and still work just fine.
I'd say the only one I've ever had & noticed the pump action bar 'bind' on any of the dozens of pump shotguns I've owned is a Marlin Model 43 (hammerless pump 12g),,and I've owned some obscure ones at that.
That was due to a combination of extreme wear (1920's Trap Gun) and as I saw it a bit of poor design. I fixed both and neither 'fix' shows on the final restoration/upgrade. Still a nice shooter.
Some have a better design as to how the action bar is attached to the pump handle metal, that's for sure. Some are just crimped together, others brazed, some use screws, some are spot welded.