That's a really hard question, as Google "Search" is more like a personal assistant than just a search engine. But I guess there is one feature which makes me ditch Google again and again: SECURITY.

The problem with Google (or rather the whole of Alphabet) is that it is too big with too many different personal products. It stores data, it connects data and it finds out stuff about you which I think is too personal. All in the pursuit of better UX. Don't get me wrong, I trust Alphabet. They use my private data to improve their products, not to sell it. But I do not want just some company (not even Google) to know me better than my very own family. There are bad people and organizations everywhere, especially the state which wants to know everything about all the people from everywhere in the world - and there are laws even Google has to abide. That's why I use proxy search engines, like Disconnect.me or DuckDuckGo; in desperate search of something better for me and the rest of the world >.>

I use DuckDuckGo already. The 0.1% of queries for which i don't get usable results, I simply append !g for a google search, !so for a stack overflow search, !w for a wikipedia search, !yt for a YouTube search, etc.

(IMO) Unfortunately, this is a space that can no longer be challenged. It would take a major corporation to attempt to build the engine and infrastructure to compete with Google; and they've tried. Microsoft attempted this with Bing and to a certain extent, they've been successful, but Google still reigns supreme.

What I would like is a search engine I can run on my own servers; tell it the domain of my site(s) and have it index them for me.

The companies in this space also fail IMO

Algolia.com (currently using) is super fast, but requires me to sync the json to their servers programmatically, nor is it self hosted.

Elastic.co (previously used) allows me to self host, but again, I must provide the inputs. It was also cumbersome to configure and maintain.

Search is very difficult to get right and people get frustrated when they can't find what they need.

Google is pretty much still at the forefront when it comes to search engine technology... But that said... I have been trying out duck duck go... And that is pretty cool. You should check it out 😄