Student or Learner
Does this meaning sound old-fashioned to your native ears?
happen on/upon something:
4. a: to meet or discover something by chance <happened upon a system that worked — Richard Corbin>
I CAN have a bit of an old-fashioned (or like a fairy tale) feel. They went for a walk, they happened upon a poor wood carver, selling his wares by the side of the road. (That whole sentence was constructed to feel like a fairy tale.)
But if you say "We were just driving home and happened upon this great little diner" it doesn't sound old fashioned to me at all, nor does it in a non-physical sense, like the example given in the dictionary -- they happened upon a solution.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.