IMHO, the drop shadow looks, to put it bluntly, quite bad. It doesn't fit in at all with the rest of RustDoc's design, which is very flat and doesn't use any dimensional design elements (such as drop shadows) anywhere else. It's pretty jarring.
As a crate maintainer, I would much rather provide multiple versions of my crate's logo for light and dark themes. I don't think that this would necessitate providing more variants as multiple themes are added — all the themes can basically be binned into "light background" and "dark background", and we can choose which logo to use based on that. If the logo looks equally good on light and dark backgrounds, the crate can just provide the same image twice. I'd much prefer this over having the drop shadow inflicted on my crates' logos.
In general, I think any solution that's based on RustDoc automatically adding something to the logo runs the risk of not looking nice with all logo designs. With that said, if having to provide multiple versions of the logo is deemed unnecessary, I think it would be much better to use a light-colored stroke or border around the logo, rather than a drop shadow — it would fit in much better with the rest of RustDoc's design language.