Good point. So our methods are similar, but yours puts the randomness in the choice of hand rather than the choice of integer. But in the (10,20) case your method gets exactly 50% and not more than 50%.
Choosing zero as a threshold deterministically does not work for all choices that Alice can make because, as you say, for some choices such as (10,20) it gives you exactly 50% chance to win. But as long as there is always a non-zero chance that you chose a threshold between Alice's numbers, then there is a > 50% chance that you win the game.