How would you build a buzz-wire game, without electronics?

Ehm... just match a DC buzzer to a battery? (or -set). The stationary wire + moving handle goes in between, acting as a switch.

Eg. 5V DC powered buzzer + 3 AA's. Or buzzer that works with 3..4V with a single Li-ion.

Use eg. stainless steel wire, even if not outside in the rain. Stays better-looking if device is stored in a damp environment like a shed. And of course battery & buzzer mounted such that no water can leak in.

Bonus: for deaf users, you could include a red LED (+series resistor) wired in parallel with the buzzer.

P.S.: this is really too simple to warrant a HN post.