No, it really isn't. Web apps and documents using the same underlying technologies doesn't mean they have to be accessed through a single frontend that provides the worst of both worlds.
I don't think that is remotely the case.
The Javascript interface isn't aware of these distinctions, though I'm not sure I want it to. Web apps like this that abuse one plain text clipboard will abuse multiple richly typed clipboards, too.
X11/Xorg has the primary and secondary selections.
MacOS has ... whatever it's got.
A key problem with this is that the feature is covert, latent, poorly discoverable, and causes unexpected behaviours. Even presumably advanced users (myself, you, other HN readers) are poorly aware of this. Imagine trying to explain to your nontechnical Aunt Tilly or Uncle Kamlesh about these "different clipboards which treat what you've copied differently and access through this funky interface"?
The browser is no longer just a document viewer... That ship has sailed, and overall it is a good thing.
We can mitigate the risk of clipboard hijacking without burning down the house. By the way, I would guess that this is a minor risk in the grand scheme of things, as it seems that the worst risks are for technical individuals people copying programmatic commands (e.g. software engineers). For others, it is a real yet minor annoyance.
Perhaps there could be an opt-in setting for allowing a site to modify the default content that is copied from a selection?