How much amnesia are we talking about? Google's core feature of being the best web search engine fits into your "advertise a bit and rebuild the customer base" criterion imo.
If google only had 10% of the search traffic it wouldn't just be smaller, it would be as irrelevant as Bing. So they would somehow need to advertise their way into a dominant position again, which is much harder than advertising just to bootstrap and aquire some custoners.
