Emergency Room staff are perfectly capable of putting 200+ items on a physical board. Not writing tasks down because it's too time consuming doesn't result in a more manageable workload of tasks, it results in people trying to remember and forgetting.
Sorry, but I don't think that's correct.
One of the big benefits of a physical kanban board is that the limited space means people only write down the stuff they really care to keep track of. For me it has never resulted in people forgetting anything important. It means they don't write down the unimportant stuff.
It's possible that some people would write down a lot of trivia or fantasy features, especially to start. The best response to that is to let them write the cards and then sort them according to actual priorities. But I've never seen anybody persist in that behavior very long. If they do, I think it's a sign of organization problems that tools can at best mask, never fix.
I think this can also be true of virtual kanban boards (e.g., GitHub's kanban view) if you keep people focused on the kanban view. Then they learn to focus on what's being worked on and the near-term to-do list. You can have a backlog column and let people fill it up as much as they want, but as long as you groom the top 20 cards or so to be your actual current priorities, people eventually adapt.