Wow, yeah, this is weird! For what it's worth, your hreflang tags seem fine, so I'd be surprised if there are canonical issues. And Google has indexed your home page.
Here's what I'd dig into: does Google think visitors want to land on your /search/ page before your homepage? I work on a site that uses a search page as a paid and organic search landing page, and it has the best conversion rate of any page on our site. Is it possible that Google thinks your homepage offers a bad user experience, and your /search/ page offers a better one, so it's picking what it thinks is best?
To see if that could be the case, I'd look into:
- What's the bounce rate of the home vs /search/ page?
- What's the time spent on page (if you can measure that properly) of each?
- How long do the pages take to load?
- How fresh is the content on the homepage vs search? (I assume the account page is the most boring.)
If this is the case, you may want to consider letting visitors search, then requiring a registration.
Good luck! And let us know what you find out!
Kristina