How Powerful was Gandalf the Grey?

[Galadriel55]

Originally Posted by R.R.J Tolkien

Gandalf really 'died', and was changed: for that seems to me the only real cheating, to represent anything that can be called 'death' as making no difference.
The Letters of J.R.R. Tolkien 156: To Robert Murray, SJ (draft). November 1954

The ‘wizards’, as such, had failed; or if you like: the crisis had become too grave and needed an enhancement of power. So Gandalf sacrificed himself, was accepted, and enhanced, and returned. 'Yes, that was the name. I was Gandalf.’ Of course he remains similar in personality and idiosyncrasy, but both his wisdom and power are much greater.
The Letters of J.R.R. Tolkien 156: To Robert Murray, SJ (draft). November 1954

I have not seen these quotes before, and I find them hard to reconcile with the rest of the philosophy of Middle-earth. I can understand Gandalf saying that Gandalf died, that Gandalf "was", implying that he is someone else - because he was indeed born anew, a reincarnation. But to imply that the fea was changed in the process - that I find hard to imagine. Things don't get re-created in Middle-earth because they failed to meet expectations or proved to be a disappointment, not in their essence. This idea goes against what feels right, Tolkien's own words or no. I just can't wrap my head around this. How do you picture this yourself? How do you reconcile this idea with everything else? I am willing to be persuaded but it needs persuasion.