The way our PM did this was that he manually updated all of the tasks himself. Once the task tracking system started working for the engineers--and this could take months--they started happily updating it.
Another strategy our PM employed was organizing stand-up around the task board. He'd specifically ask for feedback on tasks rather than round-robin asking for updates. He said the phrase, "if it isn't a task it isn't work," while also happily creating tasks for any outstanding activities unaccounted for at the end of stand-up.
So I'd recommend this strategy of the PM making task tracking work for the team before having the team make task tracking work for the PM.
Forget the official methodology, talk to your team and tell them you realize the process isn't working and ask them what would be a good way to give the stakeholders the updates they need without imposing undue work on the ICs.
Sometimes a simple and well-tolerated kanban that people update as needed is more effective than rigid daily rituals hated by everyone.
They are delivering the work, you need to understand why they don't have time to update tasks and why they don't see it as a priority.
Your goal should be to understand their position, not force them to do something that they see as low priority.
Successful projects don't come from engineers being forced to do Agile ceremonies.
If you're going to make me sit through a daily meeting where everyone goes in a circle saying all that, and most of it is "Still the same as yesterday", I'm going to resent it for wasting time.
I mean, stand-ups are whatever, 15 minutes isn't the end of the world. It's the additional combination of all the other ceremonies and endless meetings that cumulatively add up to way too much wasted time and interruption of flow.
Agile demands way too much time and breaks of focus just to provide basic details. It results in situations where X team members are waiting and twiddling their thumbs listening to the updates from X-1 other team members just so that one manager can learn what the team is up to. Not just in stand-up but every other planning ceremony and retro.
Simple async updates that happen when progress actually changes are much less demanding. Perhaps that's worth a shot?
You can force people to adopt Agile with enough pressure, but that won't magically improve transparency, productivity, or morale. It's a false promise that really just teaches engineers to optimize for the aesthetics of work (overestimating points and difficulty, shuffling tickets around, coming up with BS insights for retros, etc.) to LOOK more productive instead of actually doing the work.
A Jira query is easy enough to set up for "stories that have changed status in the past day." Check daily.
If stories aren't moving, maybe team is collaborating around current issues. That's for certain if a new deployment hasn't made it into staging.
Not sure if it works over a longer period, but that was one of the workarounds for too many Done stories to catch up on from a post-sprint perspective.