Hello and welcome Michelle. I don't have any advice to offer in choosing between TAG and CAG, having rehomed a TAG a year ago and not having had a CAG. Many of the forum members take their birds on day trips and outings with harnesses or in pack-o-birds. The classroom is another story. I'd say it depends a lot on your bird's individual personality and the way they are socialized. From what I've read and the other grey owner's I know, most grey's (TAG or CAG) aren't very comfortable in groups of people they don't know and unfamiliar situations. As Baileyspapa mentioned, people will just stick their hands right up to them not realizing they could draw back a bloody stub. I'd really be concerned that one of your students might get too close. Most greys aren't tolerant of people they don't know getting in their space and will bite, fast and hard, especially if frightened.
Many of us work all day and our birds are fine at home with plenty of toys, forage items etc. Many of us also leave the radio or tv on for them (not sure how much that helps and Timber has never told me what he thinks of the tv As long as you have a routine (they know when you are leaving and coming home) and have adequate out of cage and interaction time when you are home all will be well.
I've never seen anyone post anything about gender being a consideration with a grey. In fact, they are so similar in appearance and actions that they have to be sexed by dna. Outside of a bird laying an egg, you'd never know if you had a boy or a girl so I don't think that would be a factor in your decision.
Good luck in your hunt and keep us posted