Why don't Ikaros & Tomoki use more the Transport Cards to solve their problems?
Well, I think many can agree that SoraOto is a series which displays a comedy side and a serious side, and sometimes (many times actually) the comedic parts, like the first episode for example, serves, in the best case, for character development and, in the worst case, just for laughs. Most of the information thrown at us in the comedy parts aren't really used on the serious parts, especially when it comes to plot solving. I personally think it is just the author's decision NOT to do this.

Also, I think that at the end of episode one, Tomoki learned that he shouldn't use the cards for trivial stuff, especially after everything that happened in that first episode. Maybe that was actually the author stating why everything shouldn't be solved by a deus ex machina-like device.