It really doesn't weigh that much. Most of it gets tossed into a small saddle bag.
I've found that I flat way more with 700C tires than my old MTB. Maybe it is due to more of the tire being in contact with the ground than my old knobbies or having less rubber or some such thing. With practice you can get a tire changed in no time.
I believe I was down to 10 min or less. Far easier than trying to find a store to fix it and then waiting for them. It is possible to change a flat out in even less time.