While I wouldn't want people to be searchable in any way that makes them feel uncomfortable, that wouldn't necessarily work for my purposes. What I would really want to use that kind of search for is to find if there's anyone else studying the same language that I'm studying. So if we're discouraging people from listing languages unless they've reached some sort of mastery, then that would be a negative for me. I may only be B1-ish in Icelandic but I think that my log has some useful resources for people thinking of learning the language, at least...
The ability to search for users who are studying a certain language and then reach out to them via PM is one way of achieving this and it seems that was a feature of the old HTLAL forum. Another way could be to move to a more thread-centric model (instead of user-centric) where active users who are learning particular languages can be found via specific language threads and/or their language logs.
Unlike HTLAL, given that the search feature on this forum is free and unrestricted (for the time being) for users, this is how I'm finding and compiling a mental list in my head of who is active in certain languages and who I might be able to reach out to for help. Having said that, my first preference would be to participate in specific language threads so that groups may benefit from the mistakes that I make.
I love language logs, even the discontinued ones, because they provide a fascinating glimpse into experiments and journeys that worked for people. That alone is often inspiration enough. They're even more thrilling when I see that there are recent entries and discussions in them!
Back to the original topic though, I really like the free text format for the language box, and hopefully the community can agree on a particular standard (or a choice of two or three) for the sake of conformity, easy scanning and maybe eventually user searches.
Maybe a question for the administrators - would searching by user require the purchase and maintenance of another plugin? Or is it a default feature that comes with phpBB?