Why are skirts designed the way they are?
Skirt and Brim are not only nozzle prime operations. While Brim is intended to help in adhesion by increasing the area, Skirt is a method to test some things about the bed and positioning:


If the bed is unlevel, you can see this in the skirt before the print starts. This demands to go around all of the print.
If the bed is greasy, you'll see also before the actual print starts, as the extrusion does not stick well.
It provides a visual check if the slicing is positioned well on the printer. If you for example run printers of different dimensions, the skirt shows instantly, if you loaded G-code that is too large for the bed.


The effect to reduce airdraft to the first layer is sometimes brought up, but I don't think it is relevant. However, if one would pause between skirt and print, one could use the outline to put down glue or ABS-slurry for materials that need extra adhesion.

The use of the material in a skirt is directly proportial to the width (1 or 2 lines) and the area included (as circumference is proportional to area).


  In Cura, can I change the layout of the skirt, so that it gets drawn in a convenient part of the build platform, instead of going all the way around the edge of the print?


No. Skirt is, by definition always around the whole set of parts. Just turn off Skirt and use an alternate priming, if you don't think you'll need it or find it cumbersome. Instructions for example priming operations can be found in Writing G-code : swiping at start of print