Can multi-prime RSA be used to create an abuse-resistant lawful interception mechanism?
That sorts of works, but:


Whatever decrypts will need the full private key, and will be a potential single point of failure.
The system can be secure only if each of $pq$, $qr$, and $rp$ is wide enough to be hard to factor; so if you want to be as secure a 4096-bit RSA, you need a 6144-bit public modulus.
We can achieve the same functionality by using two-factors RSA (thus reducing the size of the public modulus at equivalent security), and Shamir's secret sharing for the private key (or the seed that was used to generate the private key, with the added bonus that the shared secrets can be short enough to be keyed-in).
We can achieve better functionality using two-factors RSA and a private/public key pair per entity, with each file encrypted with secret-key cryptography like AES and a random key, shared using Shamir's secret sharing, with the secret shares RSA-encrypted. That way, the number of autorities necessary to decipher, thresold... can be chosen on a per-file basis; and most importantly, the multiple private keys never need to be brought together.