As a SWE, would taking on management responsibilities benefit future technical career?

Do you want to manage people? Do you want to manage low performers through improvement or being fired? Telling people when they aren’t getting promoted? (Of course you also get to tell them they are) Do you crave more meetings?

Mentoring more junior engineers should already be part of a senior SWE job. Being a direct manager actually makes direct mentorship more difficult because one big part of mentorship is effectively managing the mentees challenges with their manager and how to work effectively with them. When you are the manager that isn’t really possible.

What support are you going to be given? Like is their training? Or will you roll up to an amazing senior manager or director that can grow you as both an engineer and manager?

How many reports? More than 2 is going to be a significant demand on your time. Your ability to deliver on individual contributions will be impacted, your ability to learn purely technical skills may be affected by the time demands of managing and learning to do it well.

The more positive side is that working with Eng managers, and within team constraints, changes when you’ve done the job. You understand limitations, you know why odd decisions may be made. You empathize and may be able to aid managers more effectively. You can lead a team as an engineer when there’s a manager transition, leave of absence, etc. It can be very beneficial to have these skills and as you move up as an engineer, leadership will be critical, and having the perspective of direct management can be useful.

More than anything, though, focus on if you want to do this, and be clear on a trial period where you can back out. Don’t trade a job you’re good at and enjoy for one you’re bad at and hate.

As Lee mentioned, mentoring junior engineers is effectively required to be a high-performing senior engineer. It's possible to get to Staff Engineer as a more "lone wolf" individual contributor, but it's definitely more of the exception, not the rule. The vast majority of engineers going from senior -> staff do it by uplifting others around them. If you want to get better at mentorship, I recommend starting off with this: "How can I help juniors?"

For things like approving expense reports that are clearly in the engineering manager lane, I wouldn't do them if you have 0 aspiration of becoming an engineering manager. It's just a waste of time and will distract you from Senior SWE responsibilities.