You are right that sentence 2 is better, since it is people not willingness which the site is attempting to motivate.
Even so, I think it is still too verbose.
The words motivation and willingness are near synonyms, and to include both of them is unnecessary and awkward.
Surely "I hope this site motivates people to learn English" is quite sufficient, isn't it?
ell.stackexchange.com/users/3226/WS2
There's nothing grammatically incorrect with the first sentence.
Motivate may seem like an odd modifier for willingness, but it's effectively a shorter way of saying "affect a person's willingness to learn English in a positive way" and I think it works fine here.
Your corrected sentence deviates from that meaning a bit.
Instead of modifying the level of willingness, the sentence talks about modifying whether a person has or doesn't have willingness at all.
ell.stackexchange.com/users/54646/Cantalouping
The first sentence is a bit awkward because it's as though willingness itself is the thing that is being motivated.
I think your revised version is slightly better since it is more clear that the people are the ones being motivated.
ell.stackexchange.com/users/54962/Trent Bing
It's perfectly correct to say "motivate willingness", "motivate desire", or "motivate opposition".
For example: Reconstructing Education: Toward a Pedagogy of Critical Humanism (1992) says: They claimed that a strong feeling of individual self-worth was a necessary condition to motivate students' willingness to contribute to a group.
Strategies for successful classroom teaching (1998) says: Postinstructional design also can result in motivating students' willingness to participate in group activities.
The restoration mode from Milton to Dryden (1974) says: ...
which serves partly to motivate her willingness to talk with the Hind Literature of the Sturm und Drang (2003) says: Although the play does not explicitly motivate her willingness to go to the masked ball with him...
Teacher Education Yearbook XXV
(2017) says: ...
her involvement in the Peoria Counts project generated hope through success and confidence visible in the collective positivity needed to motivate her willingness to become...
Writing:
the shapes of experience (1967) says:
And so James takes the liberty of simplifying the old lady in terms of her cupidity, which, whether it was true in reality or not, was necessary to
motivate her willingness to take a lodger in the first place
ell.stackexchange.com/users/52201/DavePhD
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.
ell.stackexchange.com/users/55030/EJoshuaS - Reinstate Monica
Answer - should be comment.
The word willingness here is synonymous with with being motivated - If a person is motivated then they are willing.
In this sense the word 'willingness' is superfluous and adds nothing to the meaning of the sentence.
If you want to correct the sentence then remove the superfluous word:
I hope this site will motivate people to learn English.
ell.stackexchange.com/users/55049/user64806
This is sunjuntive sentence so the language can be a little ungrounded.
Losing 'will' may have been an unnecessary cut, however I thing willingness is the wrong word.
I have a willingness to do my taxes.
Not a passion for doing my taxes and no even an interest in doing them.
I passively try to get them done.
The original wording works aside.
Acting on a part of a person is an action that affect the whole.
ell.stackexchange.com/users/55068/Kassandra Kaplan
I saw a sentence like this on the Internet:   I hope this site will motivate people's willingness to learn English.   I thought that there were some mistakes in the sentence, so I fixed the sentence like this:   I hope this site motivates people to have the willingness to learn English.   Do you think that I fixed the sentence well?
Is "motivate people's willingness" grammatically correct?
ell.stackexchange.com
<grammaticality>
ell.stackexchange.com/questions/128838
ell.stackexchange.com/users/1257/박용현
 This kind of usage of "motivate" is uncommon but not unheard of. 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 are willing to learn it. The first sentence is grammatically correct,  but may not mean exactly what the author seems to have intended. Although the first sentence is grammatically correct, it does not convey the meaning intended by the Author.
This type of "motivate" usage is unusual, but not unheard of. The problem is that it's unclear whether the site's goal is to persuade people to learn English or to explain why some people are willing to do so. Technically, it should mean the latter (though this is unlikely to be the author's intent). Although the first sentence is grammatically correct, it does not convey the meaning intended by the Author. The revision is a step forward.