The spiral tube mag you're thinking of was used on the pump rifle Remington Model 14 and 141. The 14 1/2 used a standard tube.
The spiral was to keep the bullet points off the primer of the cartridge ahead of it in the tube so spitzer bullets could be used in the tube feed mag.
A couple of other olde hammerless pumps are the 12ga Meriden Arms, a nice trim gun that is a lot like a Rem Model 10 as far as the trigger mechanism & bolt goes, but uses a standard style carrier,,not the M10 flipper style. About 1912 or so mfg I guess. Made for a couple years. Had Krupp Steel barrel.
I have one w/an extra bbl&forearm set. Nice $70 shooter till the already cracked stock gave it up. In the rotation for repair now.
The Stevens 200 pump shotgun. Made in 20ga w/ 3" chamber only.
About the same time period and for about the same short production,,a couple years. Very light,,5.5# maybe. Have 2 of them,,a parts gun and very nice example. No 3 inchers through the gun by me but it does shoot nice with light loads.
Very complicated action.
The 520 beats them both in production anyway if 1904/05 is it's beginning.
The rarely seen Union Arms pumps started production a bit earlier I think,,maybe '02 or 03. But not really a household name/gun nor a design to be relied upon or copied IMO.
Had a chance to buy a Stevens 525 last week,,a project gun. But too many projects now as much as I like projects and 520's.
525 was a 520 w/a solid rib, import walnut stocks and nice checkering.
I think the 535 was better wood and checkering plus engraving.