I only have to be correct for my own use case (with nearly 28 years experience being myself) and google has to be correct for everyone with (at best) 18 years of search results, of which you have to back out what the user is searching for. e.g. If I look for George Washingtons birthday I could look for information about the person george washington, the actual query george washington birthday, where I might find the data wikipedia:george washington, or an event that could contain it preseidents day.
I know both what my query is, and what a successful answer to my query is (at least the format of an acceptable return). Google has to infer that a query like presidents day and then a query like george washington are related and a successful result was the resource: wikipedia, because it contained george washington's birthday.
The point is:
* these examples are contrived. However, a person can be given the tooling to organize their own data & potential data.
* a person will have insight into what they want, almost certainly better than google.
* if they are wrong/the resources hasn't been cached for them, well, the personalized engine can just simply default to google if no results exist and return them
* privacy.
edit: not only this but having every site I visit (if I wanted this and private & personally encrypted) available and searchable is hugely valuable. Google receives takedown requests, resources move, or I forgot how I received them. Caching/downloading/storing video, images and web results and having them remain private and searchable is pretty much reason enough for me to tell you to take my money.,
So people will want to protect their data & privacy, especially companies.
People will want to have a better search experience, which means that things that are on the internet that they want to see will be there. Google is probably the most monitored for takedown requests/deindexing and they control most of the video portals and certainly the window you view the interent through so they are a target to legitimate and illegitimate takedown requests.
edit: they also have built the components and outsourced them. Hadoop/Map Reduce, most of Apache's products, elasticsearch, and another google offering tensorflow can certainly be made into a search engine. Obviously not google scale, but certainly an individual.
If anything, I think people want to have more of the interesting parts of their lives be more public (in other words, they want to be "famous", whether just among their friends or on a bigger stage). They do want many things to be "private" in the sense that they don't want certain people to see certain things online. But I think we're deluding ourselves if we think they care at all about the fact that google is scanning their emails or search history, or that twitter is mining their tweets, or that snapchat is using their location data.