You've diagnosed it correctly. Objo paints an opaque surface across the entire window: the underlying Avalonia window and its content host both carry an opaque theme background, and everything you place on the window (Canvas included) renders on transparent layers above that. So when you apply an acrylic backdrop to the window handle with DWM, our own drawing paints straight over the top of it. There isn't a plugin answer here, I'm afraid. The plugin SDK deliberately doesn't expose native window handles or the Avalonia internals, so declare-based approaches end up fighting the runtime rather than working with it.
Your 884 link is a useful pointer in reverse, though. What happened there was that a Canvas clear operation briefly ran against the window's own framebuffer and erased the window pixels, making the window see-through by accident. We fixed that by confining clears to the Canvas drawing layer, but it does demonstrate that our rendering pipeline already produces per-pixel alpha. The raw capability is there. We just don't expose it deliberately.
So here's where I've landed. I'd be happy to support window opacity and per-pixel transparency for shaped windows. Both are natively supported on Windows and macOS, and on Linux they work wherever a compositor is running, which is every modern Linux desktop, so they'd be honest cross-platform Window features rather than a Windows hack. Controls would keep their own opaque backgrounds on top, which is the "window independent of its controls" effect you asked for. The acrylic/Mica part is the one thing I'd rule out: there is no cross-platform story for blur-behind materials at all (Avalonia offers them on Windows only, macOS's equivalent isn't exposed to us, and Linux has nothing standard), so a unified Window API would carry a property that only ever does something on one platform. If it's the frosted-blur look you're specifically after, I don't want to leave you waiting on that one.
I've logged this as a feature request here: https://feedback.objo.dev/feature/1712. No promises on timing, but this is exactly the sort of thing I want Objo to be able to do.