No, it does not. Modern cryptography never relies on the secrecy of the algorithm. The reason you're having trouble finding documentation is because this is basic textbook material, so it won't appear in white papers or research.
With crypto, you always assume the attacker knows the algorithm. This poses no threat to a secure crypto system; the secret key is the only knowledge prohibited to the attacker. Note that in the case where the attacker has access to the application, they may have access to the key (depending on the application design), which is a critical failure regardless of the encryption method.
In particular, regarding the comment about trade secrets: Secure crypto does not rely on intellectual property protection at all---there is no way for trade secrets, patents, copyright, etc to keep your data safe against a persistent adversary. Ideally, you should use a homomorphic library that supports your preferred algorithm to reduce the likelihood of programming errors that undermine security.
While there are some unique dangers associated with homomorphic encryption, disclosure of the algorithm doesn't uniquely affect them. The vulnerability to IND-CCA in that paper is worrisome, and, in practice, it requires strict limits on clients/applications. FHE (fully homomorphic encryption) is maturing rapidly, and new developments may eliminate some of these issues. Regardless, it is essential to follow the guidance of the library/framework developers---and in the case of Microsoft SEAL:
scenarios where multiple different private data owners wish to engage in collaborative computation, homomorphic encryption is probably not a reasonable solution.
(The SEAL guidance generalizes to most FHE algorithms and libraries.)
Bottom line: If you are using FHE today in accordance with best practices, disclosure of the algorithm is not a significant risk. Each algorithm offers different types/levels of security and performance, however, so choosing an algorithm requires careful consideration. Newer is generally much better, but good library/framework support takes time.