"This is complex, so it must have been designed!"
Nope. It's complex, so it must be complex. We're limited creatures. Of course we couldn't design something as complex as the DNA code at the moment, but how arrogant to assume we're at the apex of natural thought in the universe! "Something must have designed it, because we, fantastic humans who basically just came down from the trees (in geological time), can't design it." No, it just means that there are natural things in this universe that are beyond our ability to copy. With complex DNA in particular, it developed naturally over time from simpler beginnings. It's not random either. We're looking at the code that survived. Most were so poorly 'designed' they didn't. We're looking at the up to date (more complex) version of it because we've arrived late on the scene. We never even knew it existed not too long ago.
Look at the sun! An immense ball of light and heat travelling across the sky. Humans couldn't make one of those and move it, therefore Ra is pulling it along in his chariot! Some things don't change.
"This impresses me, so it must have been designed!"
Yes, how amazing that we have a planet that supports life and how wonderful stars look. Except, most of the universe does not support life (lucky roll of the dice would seem to be applicable to place our planet in the right place). Stars are 'created' when gases swirl closer together due to gravitational pull. They may look good, but they still have natural beginnings. If I thought they'd been designed, I'd be wondering why they were designed to blow up at the end of their lives and destroy surrounding planets, as our sun is going to.
The great thing about Ellie's testimony is that there is good reason to believe something more happened than just her falling to the water- the 18 hours of static. The book lists other things too - her watch showing a later time for example.
The only thing that suggests a god is human ignorance, as it always has been with every other god people have worshipped.