Good question! The process of self-publishing is quite different to trad publishing - it's not that people try to get a agent/traditionally publish and if it fails they self-publish. Self-publishing isn't a 'substandard' form of publishing, just a different business model. If you have self published and sold more than 1,000 copies that would make a publisher sit up and take notice in most places I've worked.
For the vast majority, they are just two very separate routes to market.
With self publishing, you are in charge of everything, so get all the money, but you also need to be editor, marketer, publicist, brand manager, production and possibly brand extension team if you want to have products (eg soft toys of your characters).
Some authors love and thrive off that. The benefits are that you are quicker to market, keep all the cash and control your brand.
If you are traditionally published, the pay is probably quite low, and you might be competing with huge super-brands for your campaign team's resources, when they might be focussing on their bigger brands like Walliams, Peppa Pig, the Dahl estate or Harry Potter. The benefits are that you get more input and support so it's easier to build your brand. Also parents 'trust' brands like Ladybird so they are more likely to take a chance on an author they don't know. It's easier to get sold into more bookshops if you are traditionally published, whereas nearly all self-published books are sold on Amazon, via events, and in a handful of local shops.
Children's publishing even more than adult is built on heritage brands, which are what makes the megabucks, so you can find it hard to break through against a sea of Donaldson and Walliams. However, you can capitalise in your local area by doing library visits, going to schools, and doing signings or readings in local bookshops, growing your brand gradually.