Brake master cylinders (the part under the hood that turns brake pedal motion into hydraulic pressure) have two pistons in them. The pedal pushes on one piston which pushes on brake fluid to the rear
wheel brakes. Once the rear
wheel brake pads move and start pushing on the rotors the fluid pressure starts increasing. That increased pressure exists in the entire system, including inside the master cylinder. That pressure then starts pushing on the second piston which in turn pressurizes the fluid for the front brakes. The idea is to make sure the rear brakes engage first (for stability) and to absorb any brake pad-to-rotor clearance/space/gap quickly. (this was really important for car with rear drum brakes as they always have a gap; disc brakes have much less gap)
That's the normal process. However... if there is a leak in the rear brake hydraulics then pressure does not build up in the rear system... and thus the front brakes don't get applied either. After a fair bit of brake pedal motion though the rear brake piston in the master cylinder moves far enough that it physically contacts the front piston and starts pushing on it - instead of having fluid pressure do the job. Now the front brakes work normally. It's a backup/fail-safe mechanism. This often happens when the rubber seals inside the master cylinder fail or get damaged by dirt/grit in the brake fluid. The seals have to push against the (ideally) perfectly round shape of the master cylinder internal bore. If that seal has any damage the fluid is squeezed through the hole instead of making pressure in the brakes.
Some simple tests for this condition.
* if the pedal consistently needs excess travel, after a few stop-n-go drives around the block, hold your hand NEAR (DO NOT TOUCH!) the
wheel rim spokes and openings. Feel the heat being radiated by the brakes. The fronts should feel warmer... but if the rear brakes seem to be stone cold then they are not working at all due to a problem with the master cylinder or other issues. Or, if you have one, use one of those infrared temp sensor tools to measure the brake rotor temps.
* The master cylinder bolts to the front of the power brake booster unit. In theory there is NO brake fluid in that area. So loosening the master cylinder and pulling it forward a little bit (whatever the brake fluid metal piping allows) to see if there is any wetness/fluid there is a common test. If the leak is really bad there will be signs of fluid drips on the booster at the 6 o'clock position as the fluid dribbles down. Usually the paint on the booster will be damaged as brake fluid eats car paint. When the leak is this bad the brake fluid level in the reservoir will slowly drop. When the leak first starts though you won't see a change in the reservoir level.
Depending on the design of the stability control and anti-lock brakes (ABS) a failure in that system could also result in a no-brake condition. Ideally failures in the ABS should result in a "fail passive" condition - i.e. any ABS failures cause the braking system to revert to an old style non-ABS system with normal pedal feel. However, there have been ABS system designs that can fail in a mode that basically bypasses the entire braking system. GM pickups years ago had a failure type in the metal valve assembly that "bled" brake pressure away to eliminate
wheel lockup: if that valve got jammed in the "bleed" position then all brake pressure was lost. Metal shavings were commonly found in these valves. I don't know the Genesis system well enough to know if there are any failure modes that can lead to the ABS effectively bypassing all brake pressure. I hope not though; there's no reason for such a design in this day and age.
The symptom's on your wife's car sound like classic master cylinder rear cylinder leaks to me... and that generally won't show up on a computer code.
mike c.
Brakes like a champ now, but before I changed out everything, the pedal would nearly hit the floor and if I did a panic stop the brakes felt like they were overheated and really wouldn't stop the car. Scary! 
