You have inadequate draft - either your chimney is too short (should be higher than the highest portion of the home, not likely if this is a new build) or the area of the flue is too small or both.

I would not say "most" get smoke in their home when their fireplace doors are open. Much of the country with older fireplaces, with properly built chimneys of the proper height don't have any problem w/ smoke in the home from a fire - and most don't have doors on the fireplace.

You need a good quality exosfan to solve your problem. It's make your project cheaper.

Opening the door on a fire that is going and is regulated on the air supply side will almost always puff back into the room. It is a pressure issue. If I just open the door on my wood stove it too will puff back. If, however, I open the draft regulator and get more air flowing thru the stove and increase the flow up the chimney and then slowly open the door the increased draft will minimize or eliminate the puff back. With wood, every system has a personality;)

If you haven't had the chimney swept for some time that could cause smoking too.