create transparent background on a sketch created in chatgpt

The two AI-generated images you posted look more like illustrations rather than logos, especially the more complex second image. Logos are traditionally more abstract and graphical looking; details are more simplified. Those kinds of graphics are much easier to depict in vector-format. Illustrations that have a lot of pencil-sketched style details, lots of shading and other tiny variations in detail are stuck being pixel-based images.

Regarding transparent backgrounds or removing backgrounds, elements you might create from scratch in Illustrator would be vector-based objects. Something like a tan background may be nothing more than a rectangle shape stacked under/behind other elements in the foreground. Any of those objects can be moved, changed in size and manipulated in other ways without affecting the other vector objects. They won't be "baked" into each other like layers in Photoshop being merged together.

IMHO, AI-based tools tend to create more problems than they solve. This is especially true with projects like logo/branding design. An AI bot can easily generate literal images of trees, even if it looks like the trees are hand-drawn. The AI bot will struggle with more abstract, graphical ideas. Ultimately the bot is just riffing off existing visual material upon which it was trained. AI-based tools generate pixel-based results. Even the text-to-vector tools in Illustrator start with a raster-based result and then live trace the image automatically. I imagine AI-based tools for graphics work will continue to improve, but for the time being I think they're at best only good for brain-storming ideas. They're not good at generating professional quality graphics.