Anakin Skywalker as a young child knew he had abilities that most humans didn't. He had amazing reflectives and intuition. And yet it seems as though Luke had no force powers until they started to be developed by old Ben Kenobi.
Why wasn't Luke (and Leia) also inherently accessing his force powers?
Luke was aware of his ability, but didn't have much understanding of what it meant until Kenobi trained him:
Check out Children of the Jedi; from Wookieepedia's info on Luke:
During the first days of his exile, Kenobi visited the Lars homestead every day, always trying to stay clear of Owen and Beru so that they would not see him, in order to watch Luke from afar. Later Kenobi's intervention in the family business of the Lars would mark him an unwelcome man in Owen's eyes, and, as a result, Kenobi would eventually stop watching the boy every single day. According to Skywalker, his first use of the Force was when he was roughly six and, through the Force, located a lost screwdriver that was under the couch. He was severely scolded by his uncle, with the argument that he could only have known its location because he placed it there, and afterward learned not to duplicate the stunt.
(Emphasis added)
As to Leia, apparently she also used hers, too, but it was covered up:
Even though Leia was rather advanced for her age, having learned to speak and walk before the age of one, she did not show any signs of Force potential—much to the relief of her father, who knew that being a Force-sensitive in Palpatine's Empire would be a heavy burden. However, she had inherited her real father's gift, which surfaced for the first time in 18 BBY when her caregiver Memily was about to lean on a weakened fence. Leia threw a laserball at it, alerting Memily and saving her life.
Because of the incident, Leia became a target of the Galactic Empire for the first, but certainly not the last time. Word of a child with extraordinary reflexes had reached Coruscant, and Inquisitors Ferus Olin and Hydra were sent to investigate. Olin, a former Jedi apprentice, was in fact a double agent trying to save as many Force-sensitives as possible, and had come to Alderaan on Obi-Wan Kenobi's behest. Even though the Inquisitorius was not aware of the child's gender, Kenobi feared the toddler might be Leia and wanted to turn the Empire's eye away from Alderaan. Olin covered up the incident, stating it was nothing more than a rumor, and Leia's heritage remained a secret.
(Emphasis added)
It looks like she didn't pursue it, though, until years after the movies when she did become a Jedi in the New Jedi Order. I would guess it related to having enough other things 'On her plate' in her role as Princess.
