It's technically correct, but may not mean exactly what the author seems to have intended.
This kind of usage of "motivate" is uncommon but not unheard of. I've truthfully seen this usage primarily in the sense of a "motivating example" in mathematics textbooks. A "motivating example" is an example of some concept or procedure that shows why you'd even bother with it in the first place. So, in this context, "motivate people's willingness" means "explain why you'd be willing to learn English in the first place."
The problem with this usage is that it's not clear if the purpose of the site is to make people willing to learn English or to explain why some people actually are willing to learn it. Technically, it should mean the latter (which probably isn't what the author intended).
TL;DR The first sentence is grammatically correct, but it doesn't mean what the author thinks it does. Your edit is an improvement.