Has nothing to do with the motherboard or cooler. The stock cooler is the only pushpin style.. What it has to do with is the back side of the motherboard in your case. Directly behind the cpu area is usually a cutout in the frame itself, quite evident if you remove the right side panel. This is where you will be able to slide the new cpu cooler back plate into. Some cutouts are larger and make this easy, some are rather small which makes it a pain in the ash. According to the coolers directions, you setup the bolts or nuts to match your cpu, as in lga1150 or am3+ etc. I usually fold up a chunk of cardboard to fit, duct tape it over the hole then lay the pc on its side, so the motherboard is facing up and the backplate doesn't fall or move. Then install the cooler, remove the cardboard, and you are done.
The only time it's necessary to remove the motherboard itself is if there is no cutout or if it's physically beyond ability to install the cooler properly without damage.