You've clearly thought about this, and certainly begrudging use is better than none. My first thought would have been to go with Signal, if GChat through your school doesn't work. I've had no issues with group chats on there.
"Perfect is the enemy of good" -- you're making the right decision by focusing on good mentorship offering instead of blocking on the perfect chat solution. Good luck!
A. Nobody checks Slack. In the beginning everything's great and people are engaging on slack, but after a few weeks engagement drops off.
B. The number of people without Facebook has been precisely 1. If there were more people without FB accounts, I'd 100% consider another option.
C. Begrudging use is better than no use. It's not that students are boycotting my program because they have some hatred of Slack. It's more that they're confronted with a form, which they put off and then forget to do.