Why use symmetric encryption in a protocol like UPORT?
It's multi-layered encryption. If you "layer" your encryption you gain quite some nice flexibility features which is what they have done here.

You have some secret ($A$) which you only wish to share with parties $X,Y,Z$ for which you also know the public key. So what you do is that you first pick a random public-private-keypair $R$ yourself and derive a new shared secret with each of them. Now you have three shared secrets which would mean you'd have to encrypt three different copies of your message $A$. Instead you only encrypt a key $k$ three times using the shared secrets (using DLIES) and then in turn use $k$ to grant access (by means of symmetric encryption) to $A$. This approach saves you quite a bit of bandwidth and computational power there as you only need to encrypt a large message once instead of three times.

In a graphic, this process would look like this (don't forget that $k$ is small and $A$ is potentially large):