Do gas fireplace blowers actually heat rooms more effectively?
The warmth you feel when you stand in front of the fireplace is radiant heat, which is rays of energy from the hot bright flames striking surfaces and heating them up. Warming a room or space using a single source of radiant heat is difficult because you have to wait until the objects warmed by the radiant energy start to give off their own energy in the form of convected heat.

Your "fireplace blower" tries to capture heat energy with a tube or series of tubes which are placed in the path of the radiant heat energy leaving the fireplace. The theory is that a fan blowing air through the tube(s) will allow the blown air to heat up by conductive heat transfer then diffuse into the room, assisting with heating of the general area. 


  "are these blowers any good...?"


Answer- Maybe, well made units, with a large surface area to capture radiant heat from the fire and a well designed heat exchanger and fan work pretty well but are also pretty ugly IMO.


  "make it more effective...?"


Answer- If you have installed it per manufacturer's instructions there's not much more you can do except make a bigger fire. I don't have actual data, but in my experience a wood fire is generally way hotter than an equivalent "size" natural gas fire.

P.S.- I would not "put... a plain old box fan next to the fireplace and let it blow". You have a chimney for a reason. You do not want to disturb the natural draft of your fireplace. This could introduce products of combustion, like carbon monoxide, into your living space and/or blow hot embers out onto the floor.