Why is a proxy needed in proxy re-encryption? Context

Here's the usual proxy re-encryption scenario:


Alice, with key pair $(pk_a, sk_a)$ encrypts a message $m$, producing $c = encrypt(pk_a, m)$
Alice saves $c$ on an untrusted storge provider (Dropbox, IPFS, etc..)
Bob, with the key pair $(pk_b, sk_b)$ asks Alice to access $c$
Alice generates $re_{a\rightarrow b} = rekey(sk_a, pk_b)$, which she sends to a proxy
That proxy re-encrypts $c$ in $c' = reencrypt(re_{a\rightarrow b}, c)$
Bob can now download $c'$ and obtain the plaintext $m = decrypt(sk_b, c')$


Question

Why do we need a proxy? Can't Bob re-encrypt $c$ into $c'$ by himself?
From what I've understood, the only role of the proxy is to re-encrypt $c$ and nothing more. 

In other words: can Bob and the proxy be the same person/device?

Further details

If Alice sends $re_{a\rightarrow b}$ to Bob via e-mail, or by letting it be publicly accessible (i.e. storing it on IPFS), Bob should be able to re-encrypt $c$ into $c'$ by himself, provided he has the right software to do so on his device.

What I'm also interested in is uderstanding whether there are any advantages for the re-encryption step being performed by a proxy and not by the recipient himslef. I can't think of a scenario where Bob may be forced to rely on a proxy.
If Bob has the re-encryption key from Alice, then he can re-encrypt all Alice's ciphertexts stored online and then decrypt these messages. This is not good because Alice may only want to reveal some small subset of messages to Bob but not all. Similar things happen if Alice knows Bob's re-encryption key. Furthermore, the proxy is also not fully trusted, i.e., knowing Alice and Bob's re-encryption keys is not enough for the proxy to decrypt the online ciphertexts either (because it does not know Alice and Bob's private keys). On the other hand, if the proxy and, say, Bob collude, then they can decrypt all Alice's ciphertexts.