"There is no closed-form solution of the three-body problem"- is this just a folklore?

Question

Wikipedia says,

Unlike two-body problems, no general closed-form solution exists,[1] as the resulting dynamical system is chaotic for most initial conditions, and numerical methods are generally required.

So I tried to find the proof for it. However, while there are many results in Google saying "the three-body problem is not integrable", there is no such statement- "it has no closed-form solution."

Here're some details.

According to the Picard-Vessiot theory, a field $L$ is a Liouville extension of another field $K$ if and only if, the identity component of the differential Galois group of L/K is solvable. (This fact is from theorem 6.1 of Teresa Crespo and Zbigniew Hajto, Introduction to Differential Galois Theory)

On the other hands, according to the Morales-Ramis theory, a Hamiltonian system is integrable if and only if, the identity component of the differential Galois group is abelian. (This is theorem 1 of Morales-Ruiz, Juan J.; Ramis, Jean-Pierre; Simó, Carles, Integrability of hamiltonian systems and differential Galois groups of higher variational equations)

Hence, non-existence of the closed-form solution which is algebraic over elementary formulas, would imply non-integrability(since abelianness implies solvability), but not vice versa. But as I said, there are only non-integrability results on Google...

Here are some related opinons.

There is an answer in Quora saying that, non-existence of the closed-form solution is just a folklore. I'm not sure whether this is true.

(https://www.quora.com/Why-has-the-three-body-problem-not-been-solved-What-are-the-technical-obstacles-in-solving-the-three-body-problem-Are-these-difficulties-theoretical-or-not)

In the article of Wikipedia, there is a talk, which says that non-existence of the closed-term solution was not proved. As the result, word "closed-form" was removed in n-body problem subarticle, while it remained in the main article.

(https://en.wikipedia.org/wiki/Talk:Three-body_problem#Problems_with_section_n-body_problem)

An article in Scholarpedia, "Three-body problem", only states about non-integrability, and does not say anything about non-existence of the closed-form solution.

These are all I found until now. Please let me know which one is right.