The uncomfortable answer is that updating the tool is probably no one's actual job. It's everyone's implied job, which means it loses to real work every single day, and honestly it should.
What worked for shops I've been in: stop asking everyone to keep everything current. Name one person whose job is a 15-minute weekly pass asking "what on this board is no longer true?" They don't update everything themselves. They just flag what's stale and chase down the truth on the few things that matter.
The team usually stops hating the tool once it stops being a chore everyone shares and starts being one person's short, defined task.
One thing worth checking before you change anything though: is it that they don't update it, or that they update it and nothing ever comes of the updates? Different problems. The first one is the fix above. The second one means the updates go into a void, and people quit updating tools no one reads. That one gets fixed by you, not them.