So I had this really bizarre thought...

[Legolas]

The scenario here could vary wildly. Gandalf did a lot of traveling - by foot, at that! He had a great deal to tend to during the years in which these hobbits would've been growing up. Do you mean that they were to tag along with him once they were of age? That would've been very difficult, even assuming they could keep up.

If they were raised and kept with Gandalf as he traveled, they would be more adventurous and more mentally/physically suited to handle the things they had to endure during the War of the Ring.

But what about emotionally? For me, the roots they had in the Shire's simple, innocent life spurred them on as they traveled and fought with courage and resilience. This was part of Gandalf's thought process - equipping the Shire to protect itself - and a major theme of the book that is sometimes lost on readers (and more so, viewers!) in the fantasy aesthetics. He did raise them, in a certain way. They grew tremendously throughout the war.

As Gandalf himself says to them at the end of the story:

I am not coming to the Shire. You must settle its affairs yourselves; that is what you have been trained for. Do you not yet understand? My time is over: it is no longer my task to set things to rights, nor to help folk to do so. And as for you, my dear friends, you will need no help. You are grown up now. Grown indeed very high; among the great you are, and I have no longer any fear at all for any of you.