That’s not practical in many situations, as the normalization alone may very well be more expensive than the search.
If you’re in control of all data representations in your entire stack, then yes of course, but that’s hardly ever the case and different tradeoffs are made at different times (eg storage in UTF-8 because of efficiency, but in-memory representation in UTF-32 because of speed).
That doesn't make sense; the search is doing on-the-fly normalization as part of its algorithm, so it cannot be faster than normalization alone.