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Making an existing site fast on a real phone and usable by everyone on it.
Slowness and inaccessibility are usually a handful of specific causes, not a general condition. The work is finding them by measurement and removing them one at a time.
What can realistically be gained?
It depends on the cause. On this site, removing one blocking font request took first paint from 1.4s to 0.15s.
Do we have to rebuild?
Rarely. Most of the gain is in what is loaded and when, not in a rewrite.