Jane is able to see which pages you are monitoring? Perhaps she thinks you aren’t tracking the progress of the group as a whole. That might be what she means by “help”.
I think you should ask about her prior job experience. It sounds like she’s expecting more of a micro-managing style, and isn’t seeing that from you, so she’s been stepping up (as a “team player”, even!) to provide that.
Or she might be gunning for a middle-management role. How well-fitting are schedules in the team’s work? Is anyone ever late with work that holds up what other people are doing because they have to wait for that work to be finished? Jane might think that you aren’t monitoring everyone’s work because you don’t have time, and that she can show you that she DOES have time (while still getting her work done), so that she will look good for a potential promotion to supervisor.
You could address this by telling her that micro-management is not your company’s culture, and other employees are able to perform their work without constant reminders. But this assumes your team isn’t losing time to suboptimal scheduling and that she can let go of an old mindset that may have been worn in over many years. If she insists micro-managing is necessary, you might postpone/reroute her energy from “trying to adjust” to “learning how to be a supervisor”: when she knows how to do everyone’s job, she will be able to figure out where to direct her attention.
There’s an old (likely apocryphal) story about a manager running a factory where an essential piece of machinery breaks. They call in an expert, who arrives, takes a look at it, swings a wrench, and the machine begins working again. The expert says “that’ll be $800”, and the manager says “But that only took you 5 seconds! What you did isn’t worth anywhere near $800.” So the expert explains “Anyone can hit that machine with a wrench. I took 4 years of classes learning how to know WHERE to hit it with a wrench. That’s what you’re paying me for.”
Explain to Jane that what you do (as a manager) may look simple, but it’s actually about knowing exactly where your attention – and intervention – is needed, which you know because of experience doing everything the team has done. You can’t rotate her into another role just now because then there wouldn’t be enough people to do the work she’s doing right now, but if she is willing to temporarily take on the extra responsibility of training an intern, she will have an opportunity to demonstrate her leadership capabilities and make sure the team is able to keep on-schedule while she adjusts to another role. This would show her a direction to channel her ambitions in, one more productive than she’s currently exploring. If she wants to be a leader, but doesn’t want to take on the extra responsibilities you tell her it entails, well, she might realize that’s on her and stop trying to act like she’s a supervisor.
Or she might go in a different direction and try to start a union, but I think it sounds more like she’d like to be in middle-management. Testing whether she’s willing to put in the time and effort to get there may reveal not just whether she’s motivated more by the thought of having power over others, but whether (even if her intentions are noble) she’d actually be any good at it.