While I'm all for an student. I need to point out that self-improvement actions will definitely go on the back burner when she is ready to study under us.
Responsibilities that that point.
the Waystone project
the library
the spy network
this years flavour of bush fire.
Edit
teaching.
I get the sense that it's
very dangerous to assume quest turn mechanics will be the same in next arc, compared to Mathilde's time as loremaster in the EIght Peaks. We have multiple large scale game-changers at play here, from Research Institute, to Library, to Waystone Project to confidently suggest the familiar old five-AP framework (Stirland arc ran on a somewhat different framework) will necessarily fully survive the jump to the next arc. Even Lord Magistership likely puts strains on the old framework of AP/Social actions. I suspect any move to apprentices would be embedded within the mechanics changes, and for this reason, I think it's useless to speculate the impact of AP and the latitude for personal improvement.
But if you do want my take- I suspect taking Eike as Apprentice early on doesn't mean Mathilde personally instructs her in everything, but allows Mathilde to influence what courses Eike studies within the Grey College earlier in her education, before she hits Senior Apprentice, should Eike study at the Grey College in Altdorf (which is the usual course of action). This is likely going be less teaching, but mentoring - we already know a soft framework of what classes Eike could go to, instructed by other Wizards, from earlier self-improvement actions. How much time does it take to advise and mentor someone who is studying within the context of an institution? Probably about as much time Mathilde spends pouring over her EIC correspondence, or less, realistically speaking.
By the time Eike hits Senior Apprentice, per WOG, she is at the point where she could functionally serve as an assistant to Mathilde, so the time Mathilde gives to Eike likely is balanced out by the time Eike gives under Mathilde's service.
And I do find the prospect of Mathilde mentoring someone to be
interesting, both from a character development and world-building standpoint - we've got a taste of this with Ducklings. But with the Duckling club obsolete as a mechanic, I suspect the narrative space for Mathilde as mentor opens up for an apprentice, without altering much of the current writing tax.