Very interesting and makes sense to me. Storing the defaultsort should be pretty cheap.

Actually, now that I think about it it could be kind of confusing for people searching if the defaultsort is set to something other than just a rearranging of words in the title.

Instead, maybe what we need is to enable $wgCirrusSearchPrefixSearchStartsWithAnyWord everywhere so typing "Obama" will suggest "Barack Obama"

PrefixSearchStartsWithAnyWord seems to be better option, is there already existing configuration for wgCirrusSearchPrefixSearchStartsWithAnyWord?

Is there already existing configuration for $wgCirrusSearchPrefixSearchStartsWithAnyWord?
If not is it hard to implement?

@CKoerner_WMF I think it can be a nice improvement in addition to CompletionSuggester. What do you think?

We could try but I think it will lead to weird behaviors and it's maybe not what you expect, especially when you'll type more than one word in the query:
Searching for "a high" will probably suggest all Albert Highways.
Pages with short words in the title (e.g. to be or not to be) will be hard to find, searching for "to be" will probably suggest all pages concerning Tony Bennett.
It will maybe work for single word query but will probably tend to rank articles with long titles first...

I'll try to setup a large index with this option so you could have a look.