The mentor system sucks, and here's how I would fix it.

[Aramina]

I agree with several points by the OP but mostly the main one where it's a system overseen by GMs. The talk about how this would take away from their time, that's pure speculation as we have no idea how GMs operate their daily, weekly, monthly schedules. If they want to make time and/or feel like they can, they will regardless of what we think it's going to take them away from.

It can be speculation at the same time as common sense. Either they already have enough to do and that's why they are all employed or they have time to do frivolous things, such as the OP mentioned, and as such would have likely been canned already because of them not being necessary. More time in their work day is not going to somehow magically appear, and any time spent vetting people will, by necessity, take time away from something else they would have been doing.