I understand the meaning of "勾芡", but I was not able to find the origin of this term. I am trying to learn why when "勾" and "芡" combine together, they mean velveting. Thank you in advance!
The origin of the word is from acting as a soup thickener:
勾 = 调(tiáo)制 = to mix
芡 = 芡粉 = starch powder
Starch powder has often been added to soups to thicken up the soup.
Another usage of starch powder, as you alluded to in your question, is to soften meat and make it less chewy. This is more of an extension from the above usage - as the starch powder is acting more like a coating and less like something that is being mixed in.