But, but, wait! No one has addressed the obvious question. Can I run a tap down the [plate nut] [nut plate] to make it easier to accept a machine screw?
Yes, but in doing so, you will destroy it's intended locking mechanism.
When I worked as an assembler in a machine shop we would run taps through inserts to make the screws easier to install (they were Phillips so they would cam easily) then use red (permanent) Loctite on them to keep them in place. Whether this is "aviation-grade" work I have no idea but we did it for stuff going on Navy ships.