Why does flying not have any age restrictions like driving does?
Short answer: the US government agreed with you 3 years after her record flight.  

In the FAA reauthorization act of 1996, the following clause was added to 49 U.S. Code § 44724 - Manipulation of flight controls:


  (a)Prohibition.—No pilot in command of an aircraft may allow an individual who does not hold—
  
  (1) a valid private pilots certificate issued by the Administrator of the Federal Aviation Administration under part 61 of title 14, Code of Federal Regulations; and
  (2) the appropriate medical certificate issued by the Administrator under part 67 of such title,
  to manipulate the controls of an aircraft if the pilot knows or should have known that the individual is attempting to set a record or engage in an aeronautical competition or aeronautical feat, as defined by the Administrator.


Per 14 CFR 91 (can't find the exact text right now, but I know this is true from my flight training), you cannot obtain a student pilot's license until age 14 (so, no soloing until then, let alone a long-distance trip) and a private pilot's license until age 16 (for gliders).  You can't get a private pilot's license for your typical single engine aircraft until age 18.  A sport pilot's license is possible by age 17, but is limited on the aircraft you can fly (that said, a coast-to-coast trip has been flown in a Piper J-3 Cub, which fits under the sport pilot license) and conditions you can fly in.  

Hence, it would not presently be legal to do what Vicki did in 1993.  According to a NY Times article covering another young pilot, Jessica Dubroff (aged 7 at the time of her fatally unsuccessful attempt at a coast-to-coast flight): 


  Jessica was breaking no aviation rules when she flew.  The plane had dual sets of controls, meaning that the flight instructor was legally considered in charge and had the ability to take control instantly, though the plans called for Mr. Reid to take over only in an emergency.  But within hours of the crash, Federal authorities announced that they would re-examine existing regulations on underage pilots.


This would have preceded the addition to the USC quoted above.