There have been three good answers (Lesser son, TGar, Machavity) which cover most of the ground, but I'd like to expand on one point: Gandalf's apparent disinterest in the problems of the 4th Age.
The Istari were sent to M-E to guide the people of M-E, but not to rule:
For which reason the Valar are to these kindreds [Men and Elves] rather their elders and their chieftains than their masters; and if ever in their dealings with Elves and Men the Ainur have endeavoured to force them when they would not be guided, seldom has this turned to good, howsoever good the intent.
So, the Istari
...came out of the Far West and were messengers sent to contest the power of Sauron, and to unite all those who had the will to resist him; but they were forbidden to match his power with power, or to seek to dominate Elves or Men by force and fear.
They had one specific task: To contest the power of Sauron, but only by inspiring Elves and Men, not by ruling them.
The fall of Sauron would not only mark the end of a great evil, but also the ending of Elves as a significant presence in M-E. All through the Third Age
Elves, who seldom walked in the Shire, could now be seen passing westward through the woods in the evening, passing and not returning; but they were leaving Middle-earth and were no longer concerned with its troubles.
It was part of the vision of the world seen by the Ainur before Arda was created, that eventually
...the Elves should pass into the twilight and the Dominion of Men begin.
Yet what happened during the Dominion of Men may not have been part of any vision:
And some have said that the vision ceased ere the fulfilment of the Dominion of Men and the fading of the Firstborn; wherefore, though the Music is over all, the Valar have not seen as with sight the Later Ages or the ending of the World.
It was evidently Eru Iluvatar's design to, finally, leave all Arda in the hands of Men, and He didn't even tell the Ainur his plans! It was none of their business. All the more reason for the Valar to avoid meddling directly!
Gandalf (all the Istari) were embodied and this embodiment limited them and muddied their memories of Aman and of the vision they had seen before the creation of the world. (Though what memories remained may be the source of Gandalf's amazing hunches about Gollum and Bilbo and Hobbits in general.)
Gandalf spent two millennia as a mortal, an old man (unnaturally vigorous, but still subject to the weariness of the body), limited in his power, limited to using only persuasion, restrained from taking leadership -- and yet responsible for the defeat of the most powerful being on M-E, a being who had no compunction about using domination.
'The Third Age was my age. I was the Enemy of Sauron; and my work is finished. I shall go soon. The burden must lie now upon you and your kindred.'
Gandalf was ready -- more than ready -- to return home and to regain his full being. He was ready, his appointed task was successfully completed and Eru Iluvatar Himself had decreed that Men were to be on their own in the Fourth Age.
All of Gandalf's motivations were aligned. It was time to leave.