In 言葉に甘える (kotoba ni amaeru: to accept an offer), you may recognize 甘い (amai) as “sweet” … It makes perfect sense to me that a welcome offer has a connection to sweetness. After all, English speakers refer to “sweetening the pot” or “sweetening the deal.”
One more 甘 word has logic comparable to that of an English expression:
甘言 (kangen: honeyed words, flattery) sweet + words
Just as English speakers “sweet-talk” someone with flattery, Japanese speakers use “sweet words” to get their way.
Another word for “flattery” also breaks down in an interesting fashion:
巧言 (kōgen: flattery) clever, skillful + words
