Setting aside the "Eike won't need to travel" talking point - ReImagined settles that quite well in my opinion - I see three critical issues with the notion of not trying to pass on our mastery and the arguments expressed to do it.
First and what I haven't seen pointed out, masteries are not self expression. They are not designed, they are not chosen, Mathilde did not choose her shadowsteed mastery nor any of her others. They just happened - they are at most reflections of the self, and what this means for eike is that if she is incompatible with casting that reflection, she will simply not be able to learn it. If she is and develops it, it is compatible with who she is, including her personal feelings and thoughts about horses - thoughts and feelings she can use to develop her own spells from the ground up whether we pass on a mastery or not. Eike getting Mathilde's mastery does not suppress or dictate eike's personal growth in any meaningful way, not even her ability to make horse magic.
Second - there is no guarantee at all that she develops any mastery for shadowsteed what-so-ever. Mechanically, Mathilde's knowledge may influence what kind of masteries are available should the appropriate casting roll happen to develop a mastery, but it does not change the mastery chance itself. Eike is as likely to develop a mastery for shadowsteed in specific as for Burning Shadows in specific, and that means not likely what-so-ever. Not trying to pass Mathilde's mastery is giving up that high likelihood of a mastery for a lottery ticket.
And third, most obvious, Mathilde's shadowsteed is fucking awesome. Amazing. Difficult to imagine any mastery of the spell itself that could be better. It defined much of this quest, it is of immense use to Eike's stewardship of the EIC and to any application of her role as a grey magister, and if we manage to codify it will, as chocolate12 put it, redefine what you hire the grey college for. We already hit the jackpot - now, we should make sure it does not die out with us. We cannot count on codification succeeding, and we can't count on passing on the master succeeding either even if we try. We should be taking every avenue we can to make sure we are not the only person in the grey college who can do what we do.
tl;dr
1. Teaching Eike mathilde's shadowsteed does not stunt her growth or self expression in any way. She can make her own horse magic either way, the Equestrian trait/skill gives her that. But nobody can decide to make a mastery.
2. The odds of Eike getting her own shadowsteed mastery in place of getting ours are vanishingly slim in the first place. Nobody can decide to get their own mastery, and their character traits do not make that one happens more likely, only what form it takes if it does.
3. The mastery would be extraordinarily to eike in particular, and the grey college as a whole. We can pass it on as a spell via codification if that succeeds and as a mastery to Eike who will get her own crack at codification later with her personal understanding of horses backing her up. Her chance to have it is one of the biggest perks to even having mathilde as a master in the first place.
So yeah, I want mathilde to personally teach Eike shadowsteed, and to attempt to pass on that mastery.
That probably means not having her learn shadowsteed right this moment.
Edit: I don't mean to pick any fights, but I did write this in a rush - I can see a lot of issues of tone just looking back at it briefly. There's probably worse issues to find in my rhetoric if I had the time to look... So, uh. Sorry about that. X_x
