"Here's a clue. What common machine has powered reciprocating motion and a variable feed? " Haven't a clue - a tiger saw fits the bill but why the need for reciprocating unless for the design you already worked out?.
I'd thought along the same lines, either a belt or roller feed system- actually pulled the covers off my Makita bench planer to inspect the feed roller & drive. Per Makita specs, it has a feed rate of 29.8 ft / min, which boils down to 6 inches per second. Pulled the covers to see what could be done to slow it down - it's chain/sprocket driven so feed rate can be slowed down - have to look into sourcing replacement sprockets. But it hinges on finding a bench planer that can be had affordably - actually thought about contacting service centers for the various makes (Delta, Makita, etc) and offering to buy one with bad planing head, but doubt i'd be successful.
I also thought about a belt sander (i've got an old craftsman belt sander from when i worked as a carpenter in the early 70s). It's variable, and easily adapted. Hell, the sanding belts would make perfect belts but not really sure about the belts being food grade LOL. Polymer belts are probably available.
The simplest feed set up seems to be what Teck employed on the Teck 1 - a splined dowel (and currently available as "gear rod stock" in metal or polymer). What i've got sketched out is a Teck 1 with a 4" feed channel and a similiar feed dowel in the bottom, same as the Teck 1. Since that sketch, i've moved onto the belt sander, on hinge arms, being lowered from above onto the feed channel. I'm studying it to see about adding a drive takeoff for the cutting wheel/head as well.