It is actually easier to get a childrens book published with a seperate author illustrator. I don't know 100% why but that is what we've always been told at work.
I think it is because the publishers like to be able to match up author /illustrator themselves. Sometimes the author / illustrator will be the same but it is more common to find seperate ones. Also, if the publisher really likes the drawings but hates the story or vice versa they have a harder time convincing an author to stick with them and let them pick the author/writer.
Usually an author who is also the illustrator will be first and foremost an illustrator and fairly well known.
Celebrities do have an advantage of name recognition, but often they will not be the writer & illustrator. They'll be one or the other.
As for the reasons why most big publishers require an agent submit the manuscript for you, the reason is simply overload. My company used to accept "unsolicited" manuscripts but they would get in seriously hundreds a day and most of them were really bad.
I'm not trying to belittle the work these people did, but trust me, they were really bad. It would take the editorial assistants who screen all of the manuscripts before passing them along days to get through one days worth of manuscripts. They just could not do it and since the best ones tended to come from people who had agents we changed to only accepting manuscripts submitted through agents. GOOD Agents will usually only accept you as a client if they really think you can get published.
Don't be disheartened if you are turned down 100 times before someone offers to publish you.
Regardless of genre, publishers only put out about 20-30 new books each season. Usually it is even less than that. If you've written a great crime novel about a woman detective who solves cases in a rural area it would not matter if it was the greatest novel of all time if the publisher had already filled it's crime novels slot for that season.
The same with childrens authors. There has to be a niche to fill. We get over 100 AWESOME ABC / COUNTING books for kids a season here. These books are all great and very worthy of publishing, but we can only publish so many ABC/COUNTING books per year.
A lot of people get frustrated when they are turned down, and some stop trying altogether figuring their books just are not good. But the truth is, just because you don't get published by a company does not mean they don't LIKE what you've written. They may just have no need for anything in that genre or style right now.