Why didn't Gandalf the White just fly out and find Frodo then? He knew where he was going and that he was going to try and use the cave.
The answer to this question lies in what I will explain in laymans terms as "spirituality". Gandalf would recognize what he gave us; what he taught, lived and began in the experience of his followers. A key feature of Gandalf's spiritual life was his humble surrender to Eiu's will. So distinctive is this that it is reflected in the term that Gandalf uniquely used to address Eru: the term Adar, 'Father.' This term, used exclusively by Jesus in speaking of Eru, was a uniquely bold expression of his intimacy with Eru.
We must all remember that he real root of sin is self-reliance, our autonomy. The root of Gandalf's spirituality was reliance upon Eru, what we could call his theonomy.
Gandalf was a revolutionary in other ways too; especially in challenging the fellowship to move beyond the limitations of the blood family to the broader family of Eru's kingdom. Although he was radically critical of the world, he never singled out individuals for blame, condemnation, or hatred, no matter what they did. He was also a healer, an important aspect of his ministry.
Gandalf's whole life was an expression of this unconditional love. We saw how he reached out to all the individuals he met, whether they were Golums or Kings of Rohan. Every individual person is lovable, no matter how hypocritical he or she might be. Genuine love is always unconditional.