When - not if - someone finds an implementation bug in either the browser or the USB hardware, we will see exploits far worse than keylogging passwords. The bug could be in any USB device, not just the subset you're thinking about. Do you really want to trust your host-controller, every USB device, and the browser interface to be bug free? Are you sure there are no subtle non-bug interactions between devices? Have you even seen how USB devices are designed?
Limitations like "no HID devices" are how it's supposed to work, which is different from how it will actually work.
edit:
Also, did everyone forget about BadUSB?
Now, if you have a specific argument to make against my statement, then please share it. I have direct experience with the subject here, and ideally can provide context to address such arguments or clear up confusion. However, I really don't think it's productive to respond with non sequitur arguments and personal attacks.
So, the scenario you seem to be implying would require coordination between a malicious WebUSB site and a malicious device. While I can't claim that it would be impossible, it sure seems to approach de minimis if only given the extent of user interactions and preconditions.
You suggested in another comment that you had some prior background in this area, are you involved in the development of this new web API?
Justin Schuh is a Chrome security engineer and is one of the most knowledgeable people on the planet when it comes to browser security. He knows what he's talking about more than virtually anyone else in this thread.