Learn English – Term for offering something just out of reach

The definition shown in your dictionary is unnecessarily narrow, which has led to your confusion. More broadly, "to reach out" means to initiate contact with someone, with the usual implication that the contact is helpful or beneficial.

For that reason, either the helper or the person requesting help can be said to "reach out" to the other. The only stipulation is that the subject of "reach out" is the one who initiates the relationship. If you ask someone for help, then it is correct to say that you reached out for help from them; but if they offered help without you asking, then they reached out to help you.

(Note also this difference: you reached out for help, they reached out to help.)

I can't think of a way to express that without borrowing from literature, so I would probably use 'Holy Grail'. When referring to something other than a girl, it'd probably be 'White Whale'.