I don't think that's it; the usual stereotypes about Hitler and Nazis is that they were brutal, evil, and demanding, which in fact they were, not that they were silly, ridiculous, and goofy. If German sounds funny to English speakers, it's in spite of WWII associations, not because of them.
I've thought about this question a lot, and I think the answer comes from the history of English as a creole (or nearly so), consisting of a Germanic substrate being gradually displaced by a Romance prestige dialect, as the nobility all spoke dialects of Old French after the Norman Conquest. Moreover, even after that period, French was the language of diplomacy, while Latin was the language of academia and, until Henry VIII, the Church. Newton published Principia Mathematica in Latin, as was the well-established practice, and for generations studying at Harvard required learning Latin (and Greek) first. English's propensity for accepting loanwords rather than calquing them as is usual in Chinese and German has given us a large vocabulary of Latin words for use in formal contexts. New German loanwords, bu contrast, have largely come in through Yiddish, a language of desperately poor immigrants: schmuck, for example.
So it's common to have synonym pairs in which the Germanic term is informal or vulgar, while the Romance term is a formal term, sometimes an inkhorn word. Sour:acid, stuff:material, fuck:copulate, piss:urinate, cunt:vagina, cock:penis, prick:penis, shit:defecate, want:desire, fart:flatulence, balls:testicles, turd:excrement, everyday:quotidian, men:personnel, manly:virile, worldly:mundane, motherly:maternal, house:residence, big:grand, night:nocturnal, twilight:crepuscular, ass:posterior, better:ameliorate, schmuck:prepuce, water:aquatic, water:irrigate, king:monarch, armpit:axilla, cow:bovine, dog:canine, spit:saliva, rot:decay, whore:prostitute, tit:mammary, young:immature, worm:larva, enough:sufficient, grow:develop, sick:infirm, eye:ocular, think:cogitate, reckon:calculate, and so on. Pairs in the other direction are so rare I can't think of one, though I'm sure some must exist. There are cases in this list where a formal Germanic word exists, such as "breast" and "buttocks", but I can't think of a more informal Latinate synonym in those cases.
"Shit, piss, fuck, cunt, cocksucker, motherfucker, and tits," which got Carlin arrested in Milwaukee, is Germanic from beginning to end, even "suck". So is Lenny Bruce's list, even though "ass" and "balls" have cognates in Romance languages.
All the most taboo words in English except "nigger" are Germanic, and the taboo on "nigger" is recent enough that it's shaped by quite different history—but note that English speakers, to convert the Spanish negro "naygro" into a deprecating term, assimilated it to a more typically Germanic phonetic structure, ending it in a syllabic coda that is common in German and prohibited in French, Spanish, and Italian.
(To be fair, "twat" is another possible exception; nobody knows where it comes from. Although a Latin origin is improbable—literacy in Classical Rome was sufficiently broad that we know the word "landīca"—there could easily be some unattested Occitan or Sicilian word from which we get "twat", even if it sounds Germanic phonologically.)
And there's an established idiomatic way to dismiss something by reduplicating a word, the second time replacing the onset of its first syllable with the characteristically Germanic onset cluster "schm-" as a form of ridicule: "Police, schmolice!"
As a result, to Anglophone ears, German (both phonetically and in its recognizable vocables) sounds like an over-the-top vulgar version of English with words that sound a lot like "schmuckrotfart". 'What do you mean, the word for "oxygen" is "sour stuff"?'
So I suspect that German sounding silly and foolish is particular to English speakers.
