How to incentivize my Eng team to update tasks after daily standups?

There is a lot of busy work that Engineers have to do. My experience is that engineers are usually over-utilized so they naturally deprioritize things that aren't mission critical or aren't providing them value. In that regard, I think you need to prove value of task tracking to the team.

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.

Focus more on what is important to them.

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.

The code is the work. The pull request is the workface. Add automation to comment on stories that "a branch was pushed," or "a pull request was created." Then you know someone is waiting to review or it's still in progress.

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.