I made https://zetabee.com/text (which many HN users currently use) to provide a similar nested-list interface with keyboard shortcuts, copy-paste across lists, and easy read-only sharing. Before that I made http://untodos.com with 'now', 'soon', and 'whenever' lists but ZetaBee Text is a superset of untodos features so I don't use untodos anymore myself.

I haven't made a "done" feature yet because I haven't found a good way to implement it for tasks at arbitrary depths. Check out the demo: https://zetabee.com/text/demo (save is disabled for demo).

In order to get multiple lists working, you'll need to go down to the lower-right corner of each frame and select a different list to be shown in each frame. Otherwise you are just using the same list in different frames.

I'm totally with you that this UI/UX is not ideal. If Google exposed more of a Web API we could do more.

There are plenty of good web-based todo lists, what I wanted specifically was to beef up Google Tasks.

However, now that you mention it, there may not be a good multi-paned todo app out there. Someone should make it. Maybe me, maybe you! ;-)