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So with schwa-less I mean words ending in -e, -en, -er, -el don't count.

Some examples I've found: Arbeit, Armut, Heimat, Heirat Wollust, Habicht, Kranich

Though, I'm not sure whether Wollust counts as noun stem or a compound noun and Wiktionary tells me Armut and Heimat etymologically share the same suffix (with Kleinod and Einöde, which are also possible candidates for a list of such words). Looking at the etymologies of Arbeit and Heirat, it's not clear if they are also "explainable" the way the other ones are.

Is there a list of words like that somewhere? Are there any with three syllables? Ones where stress doesn't fall on the first syllable? Can you say all words of this kind can be explained by their etymology and no "true" German noun stem of Germanic origin is polysyllabic?

(I also found Atem, Jugend, Tugend, which, although not schwa-less, do interestingly sorta deviate from regular bisyllabic German noun roots.)

I found an article "The structure of the German root", by Chris Golston and Richard Wiese (published online on ResearchGate in 1998; originally published in the book Phonology and Morphology of the Germanic Languages). The analysis in this article is based on a database adapted from a list made by Wolf Dieter Ortmann (1993), who created his database based on entries in root dictionaries, mainly Augst (1975).

The article has a footnote saying "At a later stage, we plan to make the database available through the internet" (p. 68-Researchgate). I haven't been able to find it, though.

On page 75 (Researchgate), the article says that there were 131 roots in the database with two syllables like Arbeit. This excludes words where the second syllable is schwa or a schwa + resonant/syllabic resonant, but the database includes some items considered "nativized loans" (Golston and Wiese give the examples "Abenteuer ‘adventure’, add- ‘add’, Akt ‘act’, Scharlach ‘scarlet fever’") (p. 68-ResearchGate). Also, the roots seem to be of all word classes: Golston and Wiese mention "[ʔalaın]
‘alone’" as a two-syllable root.

Golston and Wiese say that there were only five three-syllable roots in the database:

The five roots in our corpus that violate the alignment constraints twice are all loans and felt to be such by most speakers: [ʔaleːgʀo] ‘allegro,’ [baldʀiaːn] (name), [ʔɛnziaːn] (name), [feːbʀuaʀ] ‘February,’ and [januaʀ] ‘January.’

(page 75-Researchgate)

Based on this, I'm fairly certain that three-syllable noun stems would be extremely marginal if not nonexistent.

Without a look at the database, it's hard to tell how many of the two-syllable examples are "nativized loans": Golston and Wiese say that the database includes 792 roots of that type, so they could potentially account for almost all of the 131 two-syllable roots in the database.

There may be a few, but I can't think of any that you haven't already mentioned.

The reasons go back to Proto-Germanic. At some unknown point (after Grimm and Verner but before Common Germanic split, so probably within 100BCE - 100CE), the original Proto-Indo-European stress disappeared. Instead, Proto-Germanic stressed all words on the first syllable, and started to reduce unstressed vowels to nothingness.

At this point, no matter how long the original Proto-Germanic root had been, it began to collapse into a monosyllable: *ēmaitijǭ > OE ǣmette > ME amte > ModE "ant". This wasn't complete by the time Proto-Germanic split apart, and didn't go all the way in all the languages—but German, English, and Norse kept running with it, and took it as far as it could go.

Thus, in these three languages, almost all native Germanic roots are monosyllables, unless this would create an illegal consonant cluster. (This exception is why we see the native word "harvest" with two syllables, because the sequence *rvst isn't valid in English—compare German Herbst.) German also reduced most vowels in endings to schwa, where English went one step further and deleted them entirely: *xagatusjǭ with its feminine ending became OHG hagzisse > German Hexe, but OE hægtesse > ME hegge > English "hag".

So while there might be a few surviving polysyllabic Germanic roots in German, like Arbeit, I wouldn't expect many. There were plenty of such roots in Proto-Germanic, but sound changes have been working tirelessly to destroy them ever since.

P.S. I was taught that an Old Norse root was always a single syllable, without exception: anything longer was a compound, and consonants were deleted all over the place to keep the syllables pronounceable. But I don't know if this is actually true or not. Knowing how unpredictable language is, I'm cautious about any sort of "always".