Another annoyance of Nano Banana (and its Pro version) is that it cannot generate transparent pixels. When it wants to, it creates a hallucinated checkerboard background that makes it worse.
Yep. Your best bet is to ask for "solid white/black background" and then feed it into something like rembg [1]. It's an extra step but it'll get you partly there.
On the OpenAI side, the gpt-image-1 model has actually had the ability to produce true alpha transparent images for a while now. Too bad quality-wise they're lagging pretty badly behind other models.