In practice, how many algorithms depend on the totient function?
I know that the security comes from the fact that the totient $\phi(n)$ is difficult to compute


Actually, we don't know that.  We know that if we can recover $\phi(n)$, we can decrypt an RSA encrypted message (given the public key), but we don't know how to go the other way; given a way to decrypt RSA encrypted message (given the ciphertext and the public key), how we might recover $\phi(n)$.  Decrypting RSA messages might be easier than recovering $\phi(n)$.

However, to address your question "what public key algorithms assume that recovering $\phi(n)$ is hard", well, recovering $\phi(n)$ is equivalent to factoring $n$ (in the sense that if you can solve one of them, you can immediately solve the other".  Hence, what you're looking for are public encryption methods that assume that factorization is difficult; I believe that list would be RSA, Rabin and Paillier (which is a public key encryption scheme with some additional homomorphic properties that can be useful at time).