"Hey, let's take a Kodak moment of this event"

Hmm, doesn't quite roll of the tongue, does it? Not the same thing.

No, because Kodak is not a free service. Xerox or Kleenex are closer, obviously, but neither of them is free to use, so there's always room to come along with something cheaper, moreover, Xerox and Kleenex at least have unbranded, one word synonyms for the commodity they produce, photocopy and tissue respectively. Google has "search the internet for", which you will never convince the average person to say, so dethroning google in "search" is at least as difficult as getting the word for "to google" changed.

Furthermore, no one can compete with Google on price, since it's free, and it's very unlikely that anyone will be able to compete on quality, unless they start disallowing adblock somehow.

The best you can hope for as a competitor is to carve out a niche of conscientious objectors, and google doesn't really care about those people either, because they are intrinsically difficult to monetize.

I'm not entirely sure "Kodak moment" was ever more than advertising, but do people still use that phrase? I know it, as a thirty-something, but I'm not sure I've ever used it.