This allows us to simplify some of the logic, and get rid of a
`gdk_seat_grab()` which might trigger unwanted popups. It's also a bit
more future-proof, as it uses the same mechanism to what is used in
GTK4 for event handling.
While we're at it, stop storing a separate `priv` pointer, which would
allow us to use `G_DECLARE_FINAL_TYPE()` in the future.
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