We're talking about two different types of search engine:
1. An information retrieval engine. This is what search engines have evolved to: Google, Bing, DDG. You get answers quick without having to go to web pages. Finding web pages is secondary for many types of query.
2. A web research engine. This is a classic search engine (ie. Google in 2001) it helps you find information located on websites not instant answers. It takes you right to the cake ignoring the frosting. This is probably easier to produce while protecting users privacy since you don't need to suss out a users location. (I'm guessing here.)
For me, when I'm acting as a directory guy looking for pages to populate categories, or when I'm looking for obscure stuff like information on webring info, I prefer #2. When I'm out and about on my phone I tend to drift more towards #1.
My bias is towards #2 because I think #1 has damaged the Web as an institution. But is that what the great unwashed masses want? I can't tell you. One of the reasons I like Mojeek is because I don't have to scroll to get to organic results, but at the same time I get easy access to Wikipedia when I need a general overview.
Algo plays in here: I keep thinking Google and Bings organic serps are too similar or are too predictable. I see the same "warhorse" sites favored in the serps for each engine.
You also need to consider the types of queries: ie. many of my queries are navigational because a company like "Honda" has so many sub websites so I'll type "honda cars" or "honda motocycles" when I just want to browse the lineup.
General on Search Engines:
IPO's are the kiss of death. Even Google, changed and not for the better, after their IPO, even though the founders retained voting control.
As a webmaster - if a search engine dose not send me traffic, it's worthless to me. There is no incentive for me to continue to make and expand web pages that are invisible to searchers and get no readers.
As a searcher - whatever gets me the information I want in the fewest clicks and minimal scrolling is generally best. But privacy is a factor there and so too is how deep I'm looking.