Secure multi-party private set intersectio: Protocols supporting MULTI-party VS supporting TWO-party Let assume party $B$ wants to receive secure multi-party computation (MPC) output. 

The are many private set intersection (PSI) protocols that support only two parties but they  cannot support multi-party. In this case, party B needs to run PSI  with  party A. Then B uses the result as the input and runs two-party PSI with party C and, so on. However, this setting leaks more information to client B, than the protocol supporting, multi-party PSI.


Example: let computation be "set intersection"

Assume party A has $S_{A}=\{1,2,3\}$

party B has $S_{B}=\{1,2\}$

party C has $S_{C}=\{2,5\}$


Party B in the protocol only supporting two-party receives:

1- $K=S_A\cap S_B=\{1,2\}$

2- $K'=S_C \cap S_B=\{2\}$

Then, it finds $K\cap K'=\{2\}$. So it learns party A has $1$ but party C does not have $1$. 

In contrast, if a protocol could support multi-party client B would only learn intersection of all sets that is $\{2\}$.

Question: What are the real-world applications (examples) of multiple-party PSI?

So I can use the examples  to show/justify that protocol supporting multi-party PSI is better (in terms of security) than those only supporting two-party.
I think that you are mixing between the terms "applications" and "implementation", let me explain.

In your question you took some secure two-party computation (2PC) and you showed that when using it in a "black-box" fashion and also each party gets the result of the computation on each pair of the inputs then the protocol leaks a lot of information that should not be leaked. This is true. But it doesn't mean that a secure multi-party computation (MPC) is useless! It just simply means that the protocol that you described (using 2PC) is leaking information and thus it is not secure!.

This multiparty set intersection is INDEED an application for MPC, it is just not implemented as you said but has some designated protocol that implements it (not necessarily through 2PC).

I hope that it clarifies the things.