It takes many clicks and server round trips to recategorize a bug, which is going to put a dent in our ability to move quickly on triage. I'd like that, when somebody changes the bug's Product dropdown, that they are immediately presented with the list of product-relevant components to choose from, rather than having to submit, round trip, new page with super-tiny menus, submit again.

Most new triagers don't know the right product anyway. The search box at https://bugzilla.mozilla.org/enter_bug.cgi?full=1 gets around that by letting you search on the component names and descriptions. Triagers should be able to use this from the bug editing screen to pick a component by searching.