I have a passing interest in linguistics[3] and so let me tell you, after intensive meditation about the topic I found that sandwich could well be cognate to German "Wecke", a bread bun, supposedly related to 'wedge'. I can't pin point "sand-" yet.

I think that would be crucial to determine the precise meaning. I'm very much against either extreme, pre- or description, but I find that expanding the sample space back to include historic meaning may yield results able to satisfy both sides.

Maybe I can still help out. We are actually talking about the club sandwich. Club sandwich could be, for all I know, which is very little about the club sandwiches history, cognate to Ger. "Klappstulle" (a 2-sandwich [to use the OP's typology] made by folding a slice over) at least the club part, and both could go all the way back to the Proto Indo European root whence glue (to stick together) compare Ger. "Kleber" (glue), "Kleie" (wheat-paste?), and perhaps Old English and Gothic "hlaif" (bread) then most likely in reference to dough, Ger. "klumpen" - 'clump', 'glob'. A club of people sticks together. The swinging and clubbing club could perhaps compare to 'key' (from Lt. clavis, from a sense nail, peg, hook), as stick, sticky and clavis, claudo show similar relations. "Klappe" means 'hatch', 'shutter', 'mouth', we say something "klappt nicht" (doesn't function), "(um)klappen" ... it's baffling that there's no good translation: The lid of box is a "Klappe", and closing it shut is "zuklappen"; A "klappe" that doesn't shut tight, flaps and makes noise, that noise is called "klappern" and from that all kinds of 'to make noise' "klappern"; Best fit would be 'clamp'. Now it depends at which time, if there was a relation at all, our dinners aligned or diverged. But, since you can't take a Hot Dog to work and eat it later--already cold--it cannot be a club sandwich, at least if folding is done primarily to stack and wrap them without making a mess of it. Well, I made a mess of it, I didn't check references on most my claims, but I'll have to wrap it up.

As a working hypothesis I suppose "sand" relates to Ger. "Kanten", the thick end of a loaf, related to "Kante" (edge), but its Greek root has no secure origin so a link to the celtic word for rim, tire giving "waybread" or "wagonbread" (cp. Ger. "Weg" - way; "Wegebrot") is not too promising. It's just, I have that image of the thick end bread as a happy hobo's ration very present from cartoons. But a better candidate would relate to butter perhaps. Or how about "Shank", cog. Ger. "Schinken" (ham), "Schenkel" (thighs), as something two sided, angled. Anyone reminded of John Connery? That's ridiculous. How about "to send for" - Ger. "gemachte Brote". I mean, there was a time and place where ... wasn't common and bread had been eaten raw or at least separately (... "Belag" - topping, coating).

[1] Wiki Commons has a picture of a ca. 8000 years old specimen in the form of a Spitzwecke https://de.wikipedia.org/wiki/Datei:Spitzweck_Reko_Oveg%C3%B...

[2] from Latin canthus (“tire, edge of a wheel”), from Ancient Greek κανθός (kanthós, “corner of the eye, tire, etc.”). The Greek is not certain, maybe from Celtic or if that seems implausible not Indo European.

[3] While I don't like anyone prescribing as if they were they majesty, I still dislike the descriptive approach that labels anything as correct that randomly comes up in the mash up of dialects. I'm especially weary of misunderstandings becoming fashionable.