It's a good start, but it's more of an illustration than a logo to be honest. It should work as a single color (white, black), at small scale and in combination with your product name.
I’ve had luck with similar things by being careful about my text prompt. Asking for tiny icon sized images also seems to clue it into the stylistic constraints of tiny icons (like what you mention).
Yes, the main usecase for DALL-E is probably for illustrations next to a story/blog. Logos are much harder to get right, and unsurprisingly DALL-E is not up to the task (yet).
It would need to be turned into a vector to scale properly but I can think of other apps that have complex logos, especially on the MacOS ecosystem. Git Tower comes to mind.
