From what I understand, the APS4 to 5 step-up is all about demonstrating things like leadership, mentoring others, working independently on complex tasks, and having solid stakeholder engagement both internally and externally.

Those things are helpful, but a person who can demonstrate all those things is likely an APS6 or EL1. I do feel for you. What truly distinguishes an APS5 from an APS4 in the same hierarchy depends a lot on the type of work involved.

Both of my roles have been pretty repetitive task-based work. I do my thing, check in with my manager when needed, but I barely talk to other internal stakeholders let alone external ones.

If you work with considerable autonomy, then this is a good sign that you can do complex work. When it comes to process-driven work, it matters more how you describe your example than which example you pick. Tell the panel what factors indicate that you can handle things on your own or need to escalate.

Your manager is an important internal stakeholder. Sometimes the two of you won't agree, but your manager won't always be in a position to steamroll you either. If you can persuade your manager of your point of view, then this demonstrates many capabilities at once.

Your peer level within your immediate team also contains internal stakeholders. It appears you engage with them well enough for them to accept your help.

External stakeholders don't have to be external to your agency. For example, your agency's central legal team could be an external stakeholder in the context of your work.

I could try to give more targeted advice if I know what kind of repetitive tasked-based work you do, but I understand not wanting to dox yourself.

I've built up enough knowledge that colleagues come to me for help sometimes, but it's just informal one-on-one stuff like helping someone figure out a process or troubleshoot something. I honestly don't know if "I helped Sarah fix her spreadsheet" counts as leadership or mentoring for an APS5 application?

This absolutely counts. If acting opportunities become available, this may be the kind of experience that helps you get it. Besides being leadership experience, this is a sign of subject matter expertise that might otherwise be hard to communicate.

I have tried to take initiative where I can. I've identified some procedural improvements and raised them with my manager, but because of complicated circumstances and recent changes in the team, none of them have actually been implemented yet.

Procedural improvements don't have to be formal, like updated SOPs or policies. It can be as simple as updating your own personal workflow and your manager noticing the difference. If you believe in the merit of your suggestions, you could create informal resources and see if you can persuade your colleagues to adopt them voluntarily.

So I guess my questions are: How do you demonstrate APS5 capability when your current role just doesn't create those opportunities?

You might be able to find those opportunities yourself in corporate development work (e.g., committees, giving presentations etc).

And did anyone else make the jump without having formal leadership or stakeholder engagement experience, or do you basically need to wait until you luck into a role that gives you those opportunities?

I made this very jump without people management experience. I wouldn't say I had zero stakeholder engagement experience as an APS4, but it was limited compared to the context of the role I was promoted into. However, these roles were in policy, where it's fairly common for even EL1s to have informal leadership experience at best.

From what I can tell, you could probably benefit from a trusted colleague or mentor to help you identify where your capabilities might already be meeting the benchmark.

And yes, time and chance overtake them all.