Jew here, I think you're misunderstanding. Conversion isn't encouraged because it's not a belief system that believes the only way is to be Jewish. Well, some more fundamentalist sects would believe that's the case, they believe they're better than everyone, even other Jews of different sects, but that's the same for any sort of extremist religious view. The 'chosen people' belief comes from what would more widely be considered to be the Old Testament (what Jews would call the Torah), so it's something that Christians and Muslims believe as well.
As for conversion, converts aren't 'sought out' because that just isn't the way. Unlike in Islam and Christianity, there is no virtue in bringing people into the fold (as in you aren't a better Jew for getting more Jews, where as Christians who bring people into Christianity are more seen as 'doing a good thing' or helping them be 'saved'). Though Jews are welcoming to people who want to become a Jew.
I'm a convert. The tradition is to turn away converts three times, so they actually have time to think about what it is that they want to do. It isn't about being 'saved'. There is no greater virtue in becoming a Jew than staying whatever it was you were before. But there is a lot of burden that comes with it potentially, and it's just the way it's done, to almost weed out those who aren't seriously committed. And it's a long process to convert, I had to take a year long course, 3 hours a week, do lots of homework, lots of meetings with and counselling by a rabbi, a bit like premarital counselling, and then even before the actual conversion, they ask you again to make sure you really understand what you're doing. It's meant to be a very educated and intentional choice, not one taken lightly or flippantly.
The other thing about Judaism is that there isn't like one set of Jewish beliefs, generally speaking. Probably a bit like Christianity. I have some very outspoken gay Christian friends and I have some very 'You will go to hell for being gay' Christian 'acquaintances' (I wouldn't call them friends). Both believe that their religion supports their attitude towards gay people in the Christian church, even though they have very opposing views. Judaism is very much the same.