Some fields were hidden and Show Aliases was turned on. After unhiding all fields and turning off aliases I can copy. It took several seconds for the [Copy] button to enable itself, perhaps 5.
I tried hiding the fields again and [Copy] remains enabled (and functional).
Copy remains enabled after exiting Pro and restarting again.
Thanks for pointing me in the right direction!
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It's a table for a polygon feature class within a feature dataset, in a file-gdb stored on network windows share.
No joins, domains, or sub-types.
I'm not sure why the strange ObjectID field name. It's not something I named deliberately. I believe the FC originally came from an AutoCAD file so it's likely something to do with that heritage.