When you quilt the layers, they shift. That is why you always have your batting and backing larger than the top. So, if you birth the quilt, you do not allow for this shifting. It can be done without the puckering, if done very carefully, with a great walking foot, and a whole lot of pinning and basting. Other than a crib size quilt, I wouldn't do it. If you have difficulty with binding, there is a third option. That is self-binding. Make your quilt in a regular way making sure your backing is at least 3" bigger on all sides. Once quilted, trim the batting to 1/2" larger than the top. Then double fold over the backing (might need to trim it a bit) to the front and machine sew it onto the front. I actually prefer to do this with all my baby quilts because it eliminates the ability of the child to chew off the bindings!

I think the difference between 'pillowcasing' and 'birthing' a quilt is that with pillowcasing you are only sewing three sides and having tofold over the entire end. With birthing, you only leave a small opening to pull it out and this helps force it to turn the seam for you. It also helps make a nie clean corner. I use this type for anything that has 4 corners (fold n' stitch wreaths, pillows, placemats, etc) because it keeps the corners nice.

I use traditional binding on quilts. I also use the 3rd option mentioned above on mug rug or small wall-hangings. Its quick and easy and no binding ends to join together.

This is how I make my self-bindings, too. but when I fold the back down the first time, I tuck some of that edge behind the batt. Then when you fold the second time, there are two layers on binding at the edge of your quilt.

Some say that you shouldn't do the self-binding because there is only one layer at the edge, but by tucking it under, there are two layers.
I think this is faster than regular binding and certainly takes much less fabric.

Whenever I "birthed" a quilt,I always made sure the batting or inner piece was about a quarter inch smaller in all directions. Then, I didn't have lumps and bumps when quilting.