A lot of agents say they don't rep picture books unless the client is both an author and illustrator. My understanding of this is that picture books aren't always the most lucrative for agents -- the author and illustrator split their share of royalties, which also reduces the agent's commission. If the same person does the text and illustrations, this issue is avoided. In my experience, many agents will also represent picture books for clients who also write for other age groups. For example, you might sign with an agent based on a middle grade manuscript but then also work with the agent on your picture books. And some agents are open to picture book authors who aren't illustrators.
Publishers are generally willing to work with one person who is both an illustrator and an author, as well -- as long as both the writing and the illustrations are up to a very high standard.
Publishers don't typically want to see partnerships between authors and illustrators, though, and neither do agents. As you said, they would rather pair the author with the illustrator. I've seen a couple of very small publishers that welcome partnerships, but this is not the way it's normally done. Unless they're planning to self-publish, authors shouldn't worry about hiring an illustrator.
If you're an illustrator and an author, you can submit your work as both, but make sure you're equally strong in both areas. If you're writing is stronger than your art, you might just want to focus on writing at first, and vice versa.
If you're a picture book author only, you can look for agent representation, but it is difficult. The good news is that some picture book publishers accept submissions directly from authors, so you can try that route, too.