Interesting that even Rama was tired. i heard one rendering here — https://www.youtube.com/watch?v=LJe939cLlKY
A quick read of the wikipedia site — https://en.wikipedia.org/wiki/Adityahridayam — is useful too.
Even the power of Sun needs to be channeled through the right warrior, in this case Rama, to defeat the darkness, the Ravana.
The question obviously is why Ravana could not get access to Aditya Hrudayam. Possibly the nature has a way of empowering the right things for right outcome !!