For someone just starting out, I'm wondering how many minutes of trot/canter should we work up to in order to successfully navigate a beginner course?
Personally, I'd work up to doing 10 minute trot sets twice a week and a day where you do two, 3 minute canters. So, for example, your weekly schedule could look something like the following -
Monday - Day off
Tuesday - 10 minute trot + hack
Wednesday - Flat lesson/school
Thursday - 3 minute canter + 2 minutes walk + 3 minutes canter
Friday - Flat lesson/school
Saturday - Jump lesson/school
Sunday - 10 minute trot + hack
Obviously you wouldn't just jump into this kind of schedule, you'd work up to it. Trot sets and canter sets are most effective, in my opinion, if you can do them over hilly terrain. If the ground is really, really hard I'd suggest just doing it in an arena to save your horse's feet and legs, or if you for some reason can't do them in an arena when there's hard ground I'd soak or pack your horse's feet afterwards. This kind of schedule would easily get your horse fit enough to run a beginner novice course, and even a novice course, though for novice I'd probably add a third 3 minute canter to the above schedule.
The idea that horses can be "too fit" and get "too high" is ridiculous to me. I'd rather have a horse a little more fit than required for the level than a horse that isn't quite fit enough.