i mow over pretty much all small sticks, or less i am near house, near roadway, or near a window or someone is around. other words if there is a chance stick getting shot out like a bullet and destroy something, i will pick up the stick. other wise pedal to the metal and keep on going.
figure time spent picking up sticks, vs sharping blades, or getting new blades for finish mower. it is easier to deal with blades. (i think i am up to 5 to 8 acres here that get mowed here on the farm) so regardless for me, blades need to be sharpen regular basis.
on larger stuff, say 1" and bigger, i will pickup, (mower just does not like that bigger stuff) and will bend the blades and tear the blades apart, and can bend and dent the deck itself. but small stuff, it is mulched, in 1 to 3 mowings.
i have thought about running a rotatory mower (bush hog) set lower to the ground. to deal with big stuff, but it leaves to large of "chunks" that finish mower does not like.
they do make kinda of finish mower blade for bush hogs. that might be heavier duty blade, that might be able to handle larger diameter branches, but never seen the blades / tried in person. just what i have seen here on the forum.
you might check out Tractor And Skid Steer Attachments For All Tractor And Skid Steer Manufacturers and look at what they have, large selection of different types of implements. and variety of rakes and like.
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ya i do not have a golf course looking yard. but i am not shooting for that look, i am happy with less maintenance, keep it green, keep it walkable, keep the dirt covered up, and ruts out, and i am happy enough. the few areas that i want better walking areas, around cabin / fire pit, paths down to lake, i will pick branches up. so i do not get someone stepping on a sharp edge of a branch when mowed, or loose footing from a stick that decides to roll under there feet on the hill down to lake.
its all about location and selective picking up small stuff, and then picking up what ever may be to large that could damage finish mower (denting, bending, both blades and deck)