It takes very little time, gathers my thoughts, and motivates me to get through what's written down. There is no greater motivater than crossing entries off the list.
"Those who have to-do lists usually manage them constantly"
Correct. By doing the work and crossing stuff off!
"half the time the tasks on the list never get done"
This isn't a fault of the list, it's the author clearly not having enough time, or not working hard enough to clear it. Either way it highlights a problem that needs fixing.
His concept of the One Thing System actually turns out to be a todo list. You just focus on the first thing on that list and don't do anything else until it is done, he even goes on to suggest making a list! :
"Make a list of possible things to do, if you like, or routine tasks that need to get done for one reason or another. "
So, don't kill your todo list, just use it properly (or better)