Claude Design is just a big opinionated prompt: https://www.lobsterpack.com/blog/claude-design-trenchcoat/. Among other things, it knows it isn't that great at drawing SVGs, so it won't try unless you force it to. For a logo, try drawing it with vanilla Claude Code as if it was a separate project: ask a "design agency" to ask you questions, answer them, then make a detailed brief of what you want to draw, then make it output an exact plan of what and where will it draw, lastly ask the chatbot to do the actual drawing using sub-agents for drawing individual components. Also add a "render it to a raster and make sure it looks right" step as well.

Claude Design is very explicitly oriented towards product UI design. It's not trying to be a product that can make you a logo.

You should try the Arrow SVG model by Quiver, should be much better at that sort of thing since it's made for that.

Anthropic has no image generation models, right?

We've got an LLM using CSS and emojis and maybe pelicans riding bikes (SVGs).