One problem I have seen with non-Google search engines is that some, including epicsearch, do not take totally seriously a search with quote marks around some words. I did a search on "Homestead Heritage" and "Yahweh" and all sorts of entries show up on the first page which are not "Homestead Heritage". I use that type of search every day, very frequently, and I have not yet found a Google replacement, but if you can help me, fine, remembering my requirement.
ProtonMail, Runbox (Norway, which I am now using, just be sure not to eat the farmed salmon), and Fastmail (USA, thus less privacy here) are examples of good non-Google mail servers. There is a forum emaildiscussions.com that is all over the comparative mail servers. If you are only using this as a Pop mailbox holding place you might end up with a $3 a month cost. However, if you really want to keep all your mail in the cloud, and you get forum mail and it is gigabytes over the years, you might have a greater cost, and you might not have the full robustness of Google search. If you limit the mailbox to personal mail, that might not be a problem, even if you do use it for long-term archiving. Points to consider.
And I actually do very little with my mail in the cloud, I have it all (even Gmail) work its way back to my "home base" on my PC. Where the living is easy. And I have worked with Eudora and TheBat! and now am considering Pandora (new Eudora Plus) email clients. For that purpose I do have a Runbox account from which I download. I am often searching for the details of correspondence and forum mail from 3, 5, 10-15 years ago.
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For research purposes I do recommend Linkman, for many of the reasons described here.
Bookmark Bliss -- Linkman by Outertech0
http://www.eve-golden.net/linkman.htm
Although I do not really use it for local files on my disk. I use it with multiple browsers, Firefox, Chrome, Palemoon, Opera, all at the same time, and have an absurdly high number of bookmarks collected over 10 years or so, and it works virtually perectly.
It will give you more privacy too, as an incidental benefit.
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Stevem