Main reasons I "shuck" my matchbooks, leaving only the match cover, with the exception of "feature" matchbooks that have graphics on the match sticks.
1) fire hazard
2) to flatten the match cover for better viewing of the outer and if present inner cover graphics
3) for more efficient storage and viewing in purpose made match cover binder pages
4) to remove the staple which can oxidize/rust and damage the match cover, especially on older pieces

This is a very common practice.

-Marshall