For those who are German or otherwise and learning English, the following two phrases are not the same (they have opposite meanings):
1. Let's ensure wise men don't stop to share their wisdom
2. Let's ensure wise men don't stop sharing their wisdom
If you say, "don't stop to do X," that means (roughly), "don't do X." In other words, #1 means the same thing as, "Let's ensure wise men don't share their wisdom." #2 is the one you want. If the amount of multiple negatives is confusing, omit them completely and say, "Let's ensure wise men keep sharing their wisdom."
Your explanation is a bit confusing, although correct. I believe this person here [1] explains it better, with more examples. The key is to “stop thinking” of these two structures as similar to each other: They're not.
This compendium of idioms from 1925 mentions it dates to the middle ages[1] – but without any details.
As an aside, I came across Proverbs 18:2, which I rather like: "A fool takes no pleasure in understanding, but only in expressing his opinion" (or, somewhat more poetic in KJV: "A fool hath no delight in understanding, but that his heart may discover itself").
Not sure about the original source though. A quick Google comes up with either German, Italian or even Lenin, with varying numbers of wise men.