When designing a crypto system we want the legitimate user to have a huge advantage over the attacker. We can increase key size as much as we want but we want legitimate users to be efficient.
In RSA the public key operation ia typically fast due to small $e$ and takes time proportional to the size of $n$.
private key operations are slower, but by using CRT we can do operations on each prime separately.
There are several possible attacks on RSA some require time proportional to smallest factor but the current best attack is NFS which though lacks rigorous analysis requires sub exponential time based on total key length.
If we focus on this attack, multi prime RSA could give us a slightly better trade off between time for running NFS and time for legitimate user to use private key.
With quantom attacks there is a proposal to use multi prime RSA to maintain a significant polynomial advantage over an adversary with a quantom computer.
As to your guess: Multi prime RSA is well known, not as well as regular RSA but plenty. Most attacks work fine against it amd specifically NFS.
I see no reason to prefer 3 primes. The number of primes does not directly make things more or less efficient as I describe cost of operations above. There is an advantage to as few primes as possible to increase the size smallest factor relative to key size. There are advantages to many primes to speed up operations with private key relative to key size.